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Номер 63, страница 110 - гдз по физике 10-11 класс задачник Гельфгат, Генденштейн
Авторы: Гельфгат И. М., Генденштейн Л. Э., Кирик Л. А.
Тип: Задачник
Издательство: Илекса
Год издания: 2008 - 2026
Уровень обучения: профильный
Цвет обложки: красный лупа, парень едет на велосипеде
ISBN: 978-5-89237-252-7
Популярные ГДЗ в 10 классе
Молекулярная физика и термодинамика. Молекулярная физика. 14. Молекулярно-кинетическая теория. Олимпиадные задачи - номер 63, страница 110.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319556 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "110" "field_page_end" => "111" "field_url" => "/10-klass/fizika/gelfgat-zadachnik/olimp-63" "field_display_title" => "63" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1074 #items: array:1 [ 0 => App\Models\Branch {#1034 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319310 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Молекулярная физика и термодинамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "104" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1038 #items: array:1 [ 0 => App\Models\Book {#1040 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1041 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1069 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1041 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1069 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319310 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Молекулярная физика и термодинамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "104" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1038} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1071} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1268 #items: array:4 [ 0 => App\Models\Branch {#1072 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319310 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Молекулярная физика и термодинамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1075 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "104" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1073 #items: array:1 [ 0 => App\Models\Book {#1076 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1077 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1079 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1082 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1101 …2} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1103 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1077 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1079 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1082 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1101 …2} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1103 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1107 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319310 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Молекулярная физика и термодинамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1075} "field_page_start" => "104" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1073} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1107} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1108 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319311 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Молекулярная физика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1109 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "104" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1110 #items: array:1 [ 0 => App\Models\Book {#1111 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1114 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1134 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1135 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1136 …2} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1137 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1138 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1139 …2} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1140 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1114 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1134 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1135 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1136 …2} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1137 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1138 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1139 …2} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1140 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1142 #items: array:1 [ 0 => App\Models\Branch {#1143 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319310 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Молекулярная физика и термодинамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1144 …2} "field_page_start" => "104" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1145 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1146 …2} ] #original: array:24 [ "id" => 1319310 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Молекулярная физика и термодинамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1144 …2} "field_page_start" => "104" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1145 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1146 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319311 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Молекулярная физика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1109} "field_page_start" => "104" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1110} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1142} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Branch {#1147 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319312 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "14. Молекулярно-кинетическая теория" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1148 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "104" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1149 #items: array:1 [ 0 => App\Models\Book {#1150 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1151 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1153 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1156 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1158 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1162 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1169 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1170 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1175 …2} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1176 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1177 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1151 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1153 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1156 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1158 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1162 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1169 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1170 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1175 …2} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1176 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1177 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1181 #items: array:1 [ 0 => App\Models\Branch {#1182 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319311 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Молекулярная физика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1183 …2} "field_page_start" => "104" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1184 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1185 …2} ] #original: array:24 [ "id" => 1319311 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Молекулярная физика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1183 …2} "field_page_start" => "104" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1184 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1185 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319312 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "14. Молекулярно-кинетическая теория" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1148} "field_page_start" => "104" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1149} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1181} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Branch {#1190 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319314 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Олимпиадные задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1191 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "108" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1192 #items: array:1 [ 0 => App\Models\Book {#1193 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1194 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1196 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1201 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1205 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1206 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1208 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1210 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1212 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1213 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1214 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1216 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1217 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1218 …2} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1219 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1220 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1221 …2} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1222 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => 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Illuminate\Database\Eloquent\Collection {#1217 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1218 …2} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1219 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1220 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1221 …2} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1222 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1224 #items: array:1 [ 0 => App\Models\Branch {#1225 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319312 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "14. 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Молекулярно-кинетическая теория" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1226 …2} "field_page_start" => "104" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1227 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1228 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319314 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Олимпиадные задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1191} "field_page_start" => "108" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1192} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1224} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1285 #items: array:2 [ 0 => App\Models\Element {#1270 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1494041 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1271 #items: array:1 [ 0 => App\Models\Edition {#1269 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5430 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1272 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "stas" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1274 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1275 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1276 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5430 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1272 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "stas" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1274 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1275 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1276 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1319556" "type" => "task" ] "text" => "<p><strong>O-63.</strong> Для улучшения теплоизоляции в колбе термоса между двойными стеклянными стенками газ находится при таком низком давлении, что длина свободного пробега молекул значительно больше расстояния между стенками колбы. Оцените давление газа в колбе термоса, если известно, что 1 кг горячей воды в нем остывает от $100 \ ^{\circ}\text{C}$ до $94 \ ^{\circ}\text{C}$ за 1 час. Между стенками колбы находится гелий.</p><p>☑ 2,2 Па.</p><p><strong><em>Решение.</em></strong> Давление газа $\text{p}$ можно определить, зная концентрацию $\text{n}$ и температуру $\text{T}$, по формуле $p = nkT$. Температуру можно принять равной $T = 97 \ ^{\circ}\text{C} = 370 \text{ K}$. Концентрацию молекул можно найти по известной скорости теплопередачи. Выберем ось OX перпендикулярной стенке баллона термоса.</p><p>На одну степень свободы молекулы приходится энергия: $\frac{m_0v_x^2}{2} = \frac{1}{2}kT$, $\overline{v_x} = \sqrt{\overline{v_x^2}} = \sqrt{\frac{kT}{m_0}}$.</p><p>Так как к одной стенке колбы летит половина молекул, то число $\nu$ ударов молекул о стенку колбы площадью $\text{S}$ в единицу времени равно: $\nu = \frac{1}{2}nS\overline{v_x} = \frac{1}{2}nS\sqrt{\frac{kT}{m_0}}$.</p><p>При каждом столкновении со стенкой кинетическая энергия молекулы становится равной среднему значению энергии теплового движения молекул стенки. При перелете от горячей стенки к холодной молекула переносит энергию: $\Delta E = E_1 - E_2 = \frac{3}{2}kT_1 - \frac{3}{2}kT_2 = \frac{3}{2}k(T_1 - T_2) = \frac{3}{2}k\Delta T_1$,</p><p>где $\Delta T_1$ — разность температур двух стенок колбы.</p><p>За время $\text{t}$ переносится количество теплоты $\text{Q}$: $Q = \Delta E\nu t = \frac{3}{2}k(T_1 - T_2)\frac{1}{2}nSt\sqrt{\frac{kT}{m_0}}$,</p><p>где $T \approx \frac{T_1 + T_2}{2}$.</p><p>По известным значениям $Q, S, T, m_0$ и $\text{t}$ можно найти концентрацию $\text{n}$: $n = \frac{4Q\sqrt{m_0}}{3\Delta T_1 St\sqrt{k^3 T}}$,</p><p>а затем найти давление $\text{p}$: $p = nkT = \frac{4Q\sqrt{m_0}T}{3S\Delta T_1 t\sqrt{k}}$,</p><p>где $Q = cm\Delta T_2$ и $\Delta T_2$ — изменение температуры воды в термосе.</p><p>$Q = 4,2 \cdot 10^3 \cdot 1,6 = 4,2 \cdot 10^4 \text{ (Дж)}$.</p><p>$m_0 = \frac{M}{N_A} = \frac{4 \cdot 10^{-3}}{6 \cdot 10^{23}} \approx 6,66 \cdot 10^{-27} \text{ (кг)}$.</p><p>Представим термос объемом 1 л в виде цилиндра радиусом $\text{r}$ и длиной $\text{l}$. $S = 2\pi rl, V = \pi r^2 l = 10^{-3} \text{ (м}^3\text{)}, S = \frac{2V}{r}$,</p><p>$S = \frac{2 \cdot 10^{-3}}{0,05} = 4 \cdot 10^{-2} \text{ (м}^2\text{)}$.</p><p>Если комнатная температура $22 \ ^{\circ}\text{C}$, то $\Delta T_1 = 75 \text{ K}$.</p><p>$p = \frac{4 \cdot 4,2 \cdot 10^4 \sqrt{6,66 \cdot 10^{-27}} \cdot 370}{3 \cdot 4 \cdot 10^{-2} \sqrt{1,38 \cdot 10^{-23}} \cdot 75 \cdot 3600} \approx 2,2 \text{ (Па)}$.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "63-1.jpg" "alt" => null "width" => "1378" "height" => 2207 "path" => "/media/fizika_10/gelfgat-z/0-olimp/63-1.webp?ts=1756251219" ] 1 => array:5 [ "name" => "63-2.jpg" "alt" => null "width" => "1327" "height" => 854 "path" => "/media/fizika_10/gelfgat-z/0-olimp/63-2.webp?ts=1756251219" ] ] ] #original: array:7 [ "id" => 1494041 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1271} "task" => array:2 [ "refs" => "1319556" "type" => "task" ] "text" => "<p><strong>O-63.</strong> Для улучшения теплоизоляции в колбе термоса между двойными стеклянными стенками газ находится при таком низком давлении, что длина свободного пробега молекул значительно больше расстояния между стенками колбы. Оцените давление газа в колбе термоса, если известно, что 1 кг горячей воды в нем остывает от $100 \ ^{\circ}\text{C}$ до $94 \ ^{\circ}\text{C}$ за 1 час. Между стенками колбы находится гелий.</p><p>☑ 2,2 Па.</p><p><strong><em>Решение.</em></strong> Давление газа $\text{p}$ можно определить, зная концентрацию $\text{n}$ и температуру $\text{T}$, по формуле $p = nkT$. Температуру можно принять равной $T = 97 \ ^{\circ}\text{C} = 370 \text{ K}$. Концентрацию молекул можно найти по известной скорости теплопередачи. Выберем ось OX перпендикулярной стенке баллона термоса.</p><p>На одну степень свободы молекулы приходится энергия: $\frac{m_0v_x^2}{2} = \frac{1}{2}kT$, $\overline{v_x} = \sqrt{\overline{v_x^2}} = \sqrt{\frac{kT}{m_0}}$.</p><p>Так как к одной стенке колбы летит половина молекул, то число $\nu$ ударов молекул о стенку колбы площадью $\text{S}$ в единицу времени равно: $\nu = \frac{1}{2}nS\overline{v_x} = \frac{1}{2}nS\sqrt{\frac{kT}{m_0}}$.</p><p>При каждом столкновении со стенкой кинетическая энергия молекулы становится равной среднему значению энергии теплового движения молекул стенки. При перелете от горячей стенки к холодной молекула переносит энергию: $\Delta E = E_1 - E_2 = \frac{3}{2}kT_1 - \frac{3}{2}kT_2 = \frac{3}{2}k(T_1 - T_2) = \frac{3}{2}k\Delta T_1$,</p><p>где $\Delta T_1$ — разность температур двух стенок колбы.</p><p>За время $\text{t}$ переносится количество теплоты $\text{Q}$: $Q = \Delta E\nu t = \frac{3}{2}k(T_1 - T_2)\frac{1}{2}nSt\sqrt{\frac{kT}{m_0}}$,</p><p>где $T \approx \frac{T_1 + T_2}{2}$.</p><p>По известным значениям $Q, S, T, m_0$ и $\text{t}$ можно найти концентрацию $\text{n}$: $n = \frac{4Q\sqrt{m_0}}{3\Delta T_1 St\sqrt{k^3 T}}$,</p><p>а затем найти давление $\text{p}$: $p = nkT = \frac{4Q\sqrt{m_0}T}{3S\Delta T_1 t\sqrt{k}}$,</p><p>где $Q = cm\Delta T_2$ и $\Delta T_2$ — изменение температуры воды в термосе.</p><p>$Q = 4,2 \cdot 10^3 \cdot 1,6 = 4,2 \cdot 10^4 \text{ (Дж)}$.</p><p>$m_0 = \frac{M}{N_A} = \frac{4 \cdot 10^{-3}}{6 \cdot 10^{23}} \approx 6,66 \cdot 10^{-27} \text{ (кг)}$.</p><p>Представим термос объемом 1 л в виде цилиндра радиусом $\text{r}$ и длиной $\text{l}$. $S = 2\pi rl, V = \pi r^2 l = 10^{-3} \text{ (м}^3\text{)}, S = \frac{2V}{r}$,</p><p>$S = \frac{2 \cdot 10^{-3}}{0,05} = 4 \cdot 10^{-2} \text{ (м}^2\text{)}$.</p><p>Если комнатная температура $22 \ ^{\circ}\text{C}$, то $\Delta T_1 = 75 \text{ K}$.</p><p>$p = \frac{4 \cdot 4,2 \cdot 10^4 \sqrt{6,66 \cdot 10^{-27}} \cdot 370}{3 \cdot 4 \cdot 10^{-2} \sqrt{1,38 \cdot 10^{-23}} \cdot 75 \cdot 3600} \approx 2,2 \text{ (Па)}$.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "63-1.jpg" "alt" => null "width" => "1378" "height" => 2207 "path" => "/media/fizika_10/gelfgat-z/0-olimp/63-1.webp?ts=1756251219" ] 1 => array:5 [ "name" => "63-2.jpg" "alt" => null "width" => "1327" "height" => 854 "path" => "/media/fizika_10/gelfgat-z/0-olimp/63-2.webp?ts=1756251219" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1277 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1691598 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1278 #items: array:1 [ 0 => App\Models\Edition {#1279 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 6138 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1280 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1282 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1283 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1284 …2} "field_content_source" => null ] #original: array:21 [ "id" => 6138 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1280 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1282 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1283 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1284 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1319556" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Масса воды, $m = 1$ кг</p><p>Начальная температура воды, $t_{1в} = 100$ °C</p><p>Конечная температура воды, $t_{2в} = 94$ °C</p><p>Время остывания, $\tau = 1$ час</p><p>Газ между стенками - гелий (He)</p><p>Удельная теплоемкость воды, $c = 4200$ Дж/(кг·К)</p><p>Молярная масса гелия, $M = 4 \cdot 10^{-3}$ кг/моль</p><p>Постоянная Авогадро, $N_A = 6.022 \cdot 10^{23}$ моль<sup>-1</sup></p><p>Постоянная Больцмана, $k = 1.38 \cdot 10^{-23}$ Дж/К</p><p><em>Оценочные данные, принимаемые для решения:</em></p><p>Объем термоса, $V \approx 1$ л</p><p>Комнатная температура (температура внешней стенки), $t_{комн} \approx 22$ °C</p><p>Радиус колбы термоса, $r \approx 5$ см</p><p>$m = 1$ кг</p><p>$\Delta t_в = 100 - 94 = 6$ °C $= 6$ К</p><p>$\tau = 1 \text{ час} = 3600$ с</p><p>$V = 1 \text{ л} = 10^{-3}$ м<sup>3</sup></p><p>$T_{2} = 22 + 273.15 = 295.15$ К (температура холодной стенки)</p><p>$r = 5 \text{ см} = 0.05$ м</p><p><strong>Найти:</strong></p><p>Давление газа, $p$.</p><p><strong>Решение:</strong></p><p>Давление газа $p$ связано с его концентрацией $n$ и температурой $T$ уравнением состояния идеального газа $p=nkT$. Концентрацию молекул $n$ можно определить из скорости теплопередачи через газ между стенками термоса.</p><p><strong>1.</strong> Сначала найдем количество теплоты $Q$, которое теряет вода при остывании. Эта теплота полностью передается через разреженный газ от внутренней стенки колбы к внешней.</p><p>$Q = c m (t_{1в} - t_{2в}) = 4200 \frac{Дж}{кг \cdot К} \cdot 1 \text{ кг} \cdot 6 \text{ К} = 25200$ Дж.</p><p><strong>2.</strong> Механизм теплопередачи в условиях, когда длина свободного пробега молекул значительно больше расстояния между стенками, заключается в переносе энергии отдельными молекулами. Молекула гелия (одноатомный газ) при упругом столкновении с горячей стенкой приобретает в среднем кинетическую энергию $\frac{3}{2}kT_1$, а при столкновении с холодной — $\frac{3}{2}kT_2$. Таким образом, за один перелет от горячей стенки к холодной молекула переносит энергию:</p><p>$\Delta E = \frac{3}{2}k(T_1 - T_2)$,</p><p>где $T_1$ – температура внутренней (горячей) стенки, а $T_2$ – температура внешней (холодной) стенки. Примем $T_1$ равной средней температуре воды, а $T_2$ – комнатной температуре.</p><p>$T_1 = \frac{100 + 94}{2} = 97$ °C = $370.15$ К.</p><p>$T_2 = 22$ °C = $295.15$ К.</p><p>Тогда разность температур стенок: $\Delta T_{ст} = T_1 - T_2 = 370.15 - 295.15 = 75$ К.</p><p><strong>3.</strong> Число молекул, ударяющихся о стенку площадью $S$ за единицу времени, можно оценить, рассмотрев движение вдоль оси, перпендикулярной стенкам. Половина молекул движется к стенке, поэтому число ударов в единицу времени $\nu = \frac{1}{2} n S \bar{v}_x$, где $\bar{v}_x$ – средняя скорость молекул вдоль этой оси. По теореме о равномерном распределении энергии по степеням свободы $\frac{1}{2}m_0 \bar{v}_x^2 = \frac{1}{2}kT$, откуда $\bar{v}_x = \sqrt{\frac{kT}{m_0}}$. Здесь $T$ – температура газа, которую оценим как среднюю между температурами стенок, $T = \frac{T_1 + T_2}{2}$, а $m_0$ – масса одной молекулы гелия.</p><p><strong>4.</strong> Общее количество теплоты $Q$, перенесенное за время $\tau$, равно произведению числа ударов о горячую стенку за это время на среднюю энергию, уносимую каждой молекулой:</p><p>$Q = \nu \cdot \tau \cdot \Delta E = \left(\frac{1}{2} n S \bar{v}_x\right) \tau \left(\frac{3}{2}k(T_1 - T_2)\right) = \frac{3}{4} n S \tau k(T_1 - T_2)\sqrt{\frac{kT}{m_0}}$</p><p><strong>5.</strong> Свяжем концентрацию $n$ с давлением $p$ через $n = p/(kT)$ и подставим в выражение для $Q$:</p><p>$Q = \frac{3}{4} \frac{p}{kT} S \tau k(T_1 - T_2)\sqrt{\frac{kT}{m_0}} = \frac{3 p S \tau (T_1 - T_2)}{4T} \sqrt{\frac{kT}{m_0}}$</p><p>Выразим давление $p$ из этого уравнения:</p><p>$p = \frac{4Q}{3 S \tau (T_1 - T_2)} \sqrt{\frac{m_0 T}{k}}$</p><p><strong>6.</strong> Подготовим численные значения для расчета:</p><p>$Q = 25200$ Дж.</p><p>Масса атома гелия: $m_0 = \frac{M}{N_A} = \frac{4 \cdot 10^{-3} \text{ кг/моль}}{6.022 \cdot 10^{23} \text{ моль}^{-1}} \approx 6.64 \cdot 10^{-27}$ кг.</p><p>Средняя температура газа: $T = \frac{370.15 + 295.15}{2} = 332.65$ К.</p><p>Площадь поверхности $S$ оценим, представив термос в виде цилиндра объемом $V=1$ л и радиусом $r=5$ см. Площадь боковой поверхности, через которую идет основная теплопередача: $S \approx 2\pi r l = 2\pi r \frac{V}{\pi r^2} = \frac{2V}{r}$.</p><p>$S = \frac{2 \cdot 10^{-3} \text{ м}^3}{0.05 \text{ м}} = 0.04$ м<sup>2</sup>.</p><p><strong>7.</strong> Произведем вычисления:</p><p>$p = \frac{4 \cdot 25200}{3 \cdot 0.04 \cdot 3600 \cdot 75} \sqrt{\frac{6.64 \cdot 10^{-27} \cdot 332.65}{1.38 \cdot 10^{-23}}} = \frac{100800}{32400} \sqrt{\frac{2.206 \cdot 10^{-24}}{1.38 \cdot 10^{-23}}} = 3.111 \cdot \sqrt{0.16} \approx 3.111 \cdot 0.4 = 1.244$ Па.</p><p><strong>Ответ:</strong> Давление газа в колбе термоса составляет примерно 1,24 Па.</p>" ] #original: array:6 [ "id" => 1691598 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1278} "task" => array:2 [ "refs" => "1319556" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Масса воды, $m = 1$ кг</p><p>Начальная температура воды, $t_{1в} = 100$ °C</p><p>Конечная температура воды, $t_{2в} = 94$ °C</p><p>Время остывания, $\tau = 1$ час</p><p>Газ между стенками - гелий (He)</p><p>Удельная теплоемкость воды, $c = 4200$ Дж/(кг·К)</p><p>Молярная масса гелия, $M = 4 \cdot 10^{-3}$ кг/моль</p><p>Постоянная Авогадро, $N_A = 6.022 \cdot 10^{23}$ моль<sup>-1</sup></p><p>Постоянная Больцмана, $k = 1.38 \cdot 10^{-23}$ Дж/К</p><p><em>Оценочные данные, принимаемые для решения:</em></p><p>Объем термоса, $V \approx 1$ л</p><p>Комнатная температура (температура внешней стенки), $t_{комн} \approx 22$ °C</p><p>Радиус колбы термоса, $r \approx 5$ см</p><p>$m = 1$ кг</p><p>$\Delta t_в = 100 - 94 = 6$ °C $= 6$ К</p><p>$\tau = 1 \text{ час} = 3600$ с</p><p>$V = 1 \text{ л} = 10^{-3}$ м<sup>3</sup></p><p>$T_{2} = 22 + 273.15 = 295.15$ К (температура холодной стенки)</p><p>$r = 5 \text{ см} = 0.05$ м</p><p><strong>Найти:</strong></p><p>Давление газа, $p$.</p><p><strong>Решение:</strong></p><p>Давление газа $p$ связано с его концентрацией $n$ и температурой $T$ уравнением состояния идеального газа $p=nkT$. Концентрацию молекул $n$ можно определить из скорости теплопередачи через газ между стенками термоса.</p><p><strong>1.</strong> Сначала найдем количество теплоты $Q$, которое теряет вода при остывании. Эта теплота полностью передается через разреженный газ от внутренней стенки колбы к внешней.</p><p>$Q = c m (t_{1в} - t_{2в}) = 4200 \frac{Дж}{кг \cdot К} \cdot 1 \text{ кг} \cdot 6 \text{ К} = 25200$ Дж.</p><p><strong>2.</strong> Механизм теплопередачи в условиях, когда длина свободного пробега молекул значительно больше расстояния между стенками, заключается в переносе энергии отдельными молекулами. Молекула гелия (одноатомный газ) при упругом столкновении с горячей стенкой приобретает в среднем кинетическую энергию $\frac{3}{2}kT_1$, а при столкновении с холодной — $\frac{3}{2}kT_2$. Таким образом, за один перелет от горячей стенки к холодной молекула переносит энергию:</p><p>$\Delta E = \frac{3}{2}k(T_1 - T_2)$,</p><p>где $T_1$ – температура внутренней (горячей) стенки, а $T_2$ – температура внешней (холодной) стенки. Примем $T_1$ равной средней температуре воды, а $T_2$ – комнатной температуре.</p><p>$T_1 = \frac{100 + 94}{2} = 97$ °C = $370.15$ К.</p><p>$T_2 = 22$ °C = $295.15$ К.</p><p>Тогда разность температур стенок: $\Delta T_{ст} = T_1 - T_2 = 370.15 - 295.15 = 75$ К.</p><p><strong>3.</strong> Число молекул, ударяющихся о стенку площадью $S$ за единицу времени, можно оценить, рассмотрев движение вдоль оси, перпендикулярной стенкам. Половина молекул движется к стенке, поэтому число ударов в единицу времени $\nu = \frac{1}{2} n S \bar{v}_x$, где $\bar{v}_x$ – средняя скорость молекул вдоль этой оси. По теореме о равномерном распределении энергии по степеням свободы $\frac{1}{2}m_0 \bar{v}_x^2 = \frac{1}{2}kT$, откуда $\bar{v}_x = \sqrt{\frac{kT}{m_0}}$. Здесь $T$ – температура газа, которую оценим как среднюю между температурами стенок, $T = \frac{T_1 + T_2}{2}$, а $m_0$ – масса одной молекулы гелия.</p><p><strong>4.</strong> Общее количество теплоты $Q$, перенесенное за время $\tau$, равно произведению числа ударов о горячую стенку за это время на среднюю энергию, уносимую каждой молекулой:</p><p>$Q = \nu \cdot \tau \cdot \Delta E = \left(\frac{1}{2} n S \bar{v}_x\right) \tau \left(\frac{3}{2}k(T_1 - T_2)\right) = \frac{3}{4} n S \tau k(T_1 - T_2)\sqrt{\frac{kT}{m_0}}$</p><p><strong>5.</strong> Свяжем концентрацию $n$ с давлением $p$ через $n = p/(kT)$ и подставим в выражение для $Q$:</p><p>$Q = \frac{3}{4} \frac{p}{kT} S \tau k(T_1 - T_2)\sqrt{\frac{kT}{m_0}} = \frac{3 p S \tau (T_1 - T_2)}{4T} \sqrt{\frac{kT}{m_0}}$</p><p>Выразим давление $p$ из этого уравнения:</p><p>$p = \frac{4Q}{3 S \tau (T_1 - T_2)} \sqrt{\frac{m_0 T}{k}}$</p><p><strong>6.</strong> Подготовим численные значения для расчета:</p><p>$Q = 25200$ Дж.</p><p>Масса атома гелия: $m_0 = \frac{M}{N_A} = \frac{4 \cdot 10^{-3} \text{ кг/моль}}{6.022 \cdot 10^{23} \text{ моль}^{-1}} \approx 6.64 \cdot 10^{-27}$ кг.</p><p>Средняя температура газа: $T = \frac{370.15 + 295.15}{2} = 332.65$ К.</p><p>Площадь поверхности $S$ оценим, представив термос в виде цилиндра объемом $V=1$ л и радиусом $r=5$ см. Площадь боковой поверхности, через которую идет основная теплопередача: $S \approx 2\pi r l = 2\pi r \frac{V}{\pi r^2} = \frac{2V}{r}$.</p><p>$S = \frac{2 \cdot 10^{-3} \text{ м}^3}{0.05 \text{ м}} = 0.04$ м<sup>2</sup>.</p><p><strong>7.</strong> Произведем вычисления:</p><p>$p = \frac{4 \cdot 25200}{3 \cdot 0.04 \cdot 3600 \cdot 75} \sqrt{\frac{6.64 \cdot 10^{-27} \cdot 332.65}{1.38 \cdot 10^{-23}}} = \frac{100800}{32400} \sqrt{\frac{2.206 \cdot 10^{-24}}{1.38 \cdot 10^{-23}}} = 3.111 \cdot \sqrt{0.16} \approx 3.111 \cdot 0.4 = 1.244$ Па.</p><p><strong>Ответ:</strong> Давление газа в колбе термоса составляет примерно 1,24 Па.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1319557" 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#hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#1296 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Term {#1297 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1298 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1299 #items: array:1 [ 0 => App\Models\Term {#1300 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1301 #items: array:1 [ 0 => App\Models\Term {#1302 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1303 #items: array:1 [ 0 => App\Models\Term {#1304 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1305 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1306 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1307 #items: array:1 [ 0 => App\Models\Term {#1308 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1309 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1310 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1311 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" 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"field_old_book" => Illuminate\Database\Eloquent\Collection {#1314 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1315 #items: array:1 [ 0 => App\Models\Term {#1316 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 556 "class_subject" => 494 ] ] #original: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1288} "field_class" => Illuminate\Database\Eloquent\Collection {#1289} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1292} "field_author" => Illuminate\Database\Eloquent\Collection {#1294} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1298} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1299} "field_country" => Illuminate\Database\Eloquent\Collection {#1301} "field_city" => Illuminate\Database\Eloquent\Collection {#1303} "field_series" => Illuminate\Database\Eloquent\Collection {#1305} "field_umk" => Illuminate\Database\Eloquent\Collection {#1306} "field_level_of_education" => 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array:1 [ 0 => array:4 [ "path" => "/media/fizika_10/gelfgat-z/covers/cover1.webp?ts=1756049816" "alt" => "" "width" => "1566" "height" => "2412" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1312} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1313} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1314} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1315} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 556 "class_subject" => 494 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] 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"field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1327 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1337 #items: array:1 [ 0 => App\Models\Element {#1336 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" 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"breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1074} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1268} "content" => Illuminate\Database\Eloquent\Collection {#1285} "next" => array:2 [ "refs" => "1319557" "type" => "task" ] "previous" => array:2 [ "refs" => "1319555" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1319} "page" => array:2 [ "refs" => "1341071" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№63 (с. 110)
Условие. №63 (с. 110)
скриншот условия
O-63. Для улучшения теплоизоляции в колбе термоса между двойными стеклянными стенками газ находится при таком низком давлении, что длина свободного пробега молекул значительно больше расстояния между стенками колбы. Оцените давление газа в колбе термоса, если известно, что 1 кг горячей воды в нем остывает от $100 \ ^{\circ}\text{C}$ до $94 \ ^{\circ}\text{C}$ за 1 час. Между стенками колбы находится гелий.
☑ 2,2 Па.
Решение. Давление газа $\text{p}$ можно определить, зная концентрацию $\text{n}$ и температуру $\text{T}$, по формуле $p = nkT$. Температуру можно принять равной $T = 97 \ ^{\circ}\text{C} = 370 \text{ K}$. Концентрацию молекул можно найти по известной скорости теплопередачи. Выберем ось OX перпендикулярной стенке баллона термоса.
На одну степень свободы молекулы приходится энергия: $\frac{m_0v_x^2}{2} = \frac{1}{2}kT$, $\overline{v_x} = \sqrt{\overline{v_x^2}} = \sqrt{\frac{kT}{m_0}}$.
Так как к одной стенке колбы летит половина молекул, то число $\nu$ ударов молекул о стенку колбы площадью $\text{S}$ в единицу времени равно: $\nu = \frac{1}{2}nS\overline{v_x} = \frac{1}{2}nS\sqrt{\frac{kT}{m_0}}$.
При каждом столкновении со стенкой кинетическая энергия молекулы становится равной среднему значению энергии теплового движения молекул стенки. При перелете от горячей стенки к холодной молекула переносит энергию: $\Delta E = E_1 - E_2 = \frac{3}{2}kT_1 - \frac{3}{2}kT_2 = \frac{3}{2}k(T_1 - T_2) = \frac{3}{2}k\Delta T_1$,
где $\Delta T_1$ — разность температур двух стенок колбы.
За время $\text{t}$ переносится количество теплоты $\text{Q}$: $Q = \Delta E\nu t = \frac{3}{2}k(T_1 - T_2)\frac{1}{2}nSt\sqrt{\frac{kT}{m_0}}$,
где $T \approx \frac{T_1 + T_2}{2}$.
По известным значениям $Q, S, T, m_0$ и $\text{t}$ можно найти концентрацию $\text{n}$: $n = \frac{4Q\sqrt{m_0}}{3\Delta T_1 St\sqrt{k^3 T}}$,
а затем найти давление $\text{p}$: $p = nkT = \frac{4Q\sqrt{m_0}T}{3S\Delta T_1 t\sqrt{k}}$,
где $Q = cm\Delta T_2$ и $\Delta T_2$ — изменение температуры воды в термосе.
$Q = 4,2 \cdot 10^3 \cdot 1,6 = 4,2 \cdot 10^4 \text{ (Дж)}$.
$m_0 = \frac{M}{N_A} = \frac{4 \cdot 10^{-3}}{6 \cdot 10^{23}} \approx 6,66 \cdot 10^{-27} \text{ (кг)}$.
Представим термос объемом 1 л в виде цилиндра радиусом $\text{r}$ и длиной $\text{l}$. $S = 2\pi rl, V = \pi r^2 l = 10^{-3} \text{ (м}^3\text{)}, S = \frac{2V}{r}$,
$S = \frac{2 \cdot 10^{-3}}{0,05} = 4 \cdot 10^{-2} \text{ (м}^2\text{)}$.
Если комнатная температура $22 \ ^{\circ}\text{C}$, то $\Delta T_1 = 75 \text{ K}$.
$p = \frac{4 \cdot 4,2 \cdot 10^4 \sqrt{6,66 \cdot 10^{-27}} \cdot 370}{3 \cdot 4 \cdot 10^{-2} \sqrt{1,38 \cdot 10^{-23}} \cdot 75 \cdot 3600} \approx 2,2 \text{ (Па)}$.
Решение. №63 (с. 110)
Дано:
Масса воды, $m = 1$ кг
Начальная температура воды, $t_{1в} = 100$ °C
Конечная температура воды, $t_{2в} = 94$ °C
Время остывания, $\tau = 1$ час
Газ между стенками - гелий (He)
Удельная теплоемкость воды, $c = 4200$ Дж/(кг·К)
Молярная масса гелия, $M = 4 \cdot 10^{-3}$ кг/моль
Постоянная Авогадро, $N_A = 6.022 \cdot 10^{23}$ моль-1
Постоянная Больцмана, $k = 1.38 \cdot 10^{-23}$ Дж/К
Оценочные данные, принимаемые для решения:
Объем термоса, $V \approx 1$ л
Комнатная температура (температура внешней стенки), $t_{комн} \approx 22$ °C
Радиус колбы термоса, $r \approx 5$ см
$m = 1$ кг
$\Delta t_в = 100 - 94 = 6$ °C $= 6$ К
$\tau = 1 \text{ час} = 3600$ с
$V = 1 \text{ л} = 10^{-3}$ м3
$T_{2} = 22 + 273.15 = 295.15$ К (температура холодной стенки)
$r = 5 \text{ см} = 0.05$ м
Найти:
Давление газа, $p$.
Решение:
Давление газа $p$ связано с его концентрацией $n$ и температурой $T$ уравнением состояния идеального газа $p=nkT$. Концентрацию молекул $n$ можно определить из скорости теплопередачи через газ между стенками термоса.
1. Сначала найдем количество теплоты $Q$, которое теряет вода при остывании. Эта теплота полностью передается через разреженный газ от внутренней стенки колбы к внешней.
$Q = c m (t_{1в} - t_{2в}) = 4200 \frac{Дж}{кг \cdot К} \cdot 1 \text{ кг} \cdot 6 \text{ К} = 25200$ Дж.
2. Механизм теплопередачи в условиях, когда длина свободного пробега молекул значительно больше расстояния между стенками, заключается в переносе энергии отдельными молекулами. Молекула гелия (одноатомный газ) при упругом столкновении с горячей стенкой приобретает в среднем кинетическую энергию $\frac{3}{2}kT_1$, а при столкновении с холодной — $\frac{3}{2}kT_2$. Таким образом, за один перелет от горячей стенки к холодной молекула переносит энергию:
$\Delta E = \frac{3}{2}k(T_1 - T_2)$,
где $T_1$ – температура внутренней (горячей) стенки, а $T_2$ – температура внешней (холодной) стенки. Примем $T_1$ равной средней температуре воды, а $T_2$ – комнатной температуре.
$T_1 = \frac{100 + 94}{2} = 97$ °C = $370.15$ К.
$T_2 = 22$ °C = $295.15$ К.
Тогда разность температур стенок: $\Delta T_{ст} = T_1 - T_2 = 370.15 - 295.15 = 75$ К.
3. Число молекул, ударяющихся о стенку площадью $S$ за единицу времени, можно оценить, рассмотрев движение вдоль оси, перпендикулярной стенкам. Половина молекул движется к стенке, поэтому число ударов в единицу времени $\nu = \frac{1}{2} n S \bar{v}_x$, где $\bar{v}_x$ – средняя скорость молекул вдоль этой оси. По теореме о равномерном распределении энергии по степеням свободы $\frac{1}{2}m_0 \bar{v}_x^2 = \frac{1}{2}kT$, откуда $\bar{v}_x = \sqrt{\frac{kT}{m_0}}$. Здесь $T$ – температура газа, которую оценим как среднюю между температурами стенок, $T = \frac{T_1 + T_2}{2}$, а $m_0$ – масса одной молекулы гелия.
4. Общее количество теплоты $Q$, перенесенное за время $\tau$, равно произведению числа ударов о горячую стенку за это время на среднюю энергию, уносимую каждой молекулой:
$Q = \nu \cdot \tau \cdot \Delta E = \left(\frac{1}{2} n S \bar{v}_x\right) \tau \left(\frac{3}{2}k(T_1 - T_2)\right) = \frac{3}{4} n S \tau k(T_1 - T_2)\sqrt{\frac{kT}{m_0}}$
5. Свяжем концентрацию $n$ с давлением $p$ через $n = p/(kT)$ и подставим в выражение для $Q$:
$Q = \frac{3}{4} \frac{p}{kT} S \tau k(T_1 - T_2)\sqrt{\frac{kT}{m_0}} = \frac{3 p S \tau (T_1 - T_2)}{4T} \sqrt{\frac{kT}{m_0}}$
Выразим давление $p$ из этого уравнения:
$p = \frac{4Q}{3 S \tau (T_1 - T_2)} \sqrt{\frac{m_0 T}{k}}$
6. Подготовим численные значения для расчета:
$Q = 25200$ Дж.
Масса атома гелия: $m_0 = \frac{M}{N_A} = \frac{4 \cdot 10^{-3} \text{ кг/моль}}{6.022 \cdot 10^{23} \text{ моль}^{-1}} \approx 6.64 \cdot 10^{-27}$ кг.
Средняя температура газа: $T = \frac{370.15 + 295.15}{2} = 332.65$ К.
Площадь поверхности $S$ оценим, представив термос в виде цилиндра объемом $V=1$ л и радиусом $r=5$ см. Площадь боковой поверхности, через которую идет основная теплопередача: $S \approx 2\pi r l = 2\pi r \frac{V}{\pi r^2} = \frac{2V}{r}$.
$S = \frac{2 \cdot 10^{-3} \text{ м}^3}{0.05 \text{ м}} = 0.04$ м2.
7. Произведем вычисления:
$p = \frac{4 \cdot 25200}{3 \cdot 0.04 \cdot 3600 \cdot 75} \sqrt{\frac{6.64 \cdot 10^{-27} \cdot 332.65}{1.38 \cdot 10^{-23}}} = \frac{100800}{32400} \sqrt{\frac{2.206 \cdot 10^{-24}}{1.38 \cdot 10^{-23}}} = 3.111 \cdot \sqrt{0.16} \approx 3.111 \cdot 0.4 = 1.244$ Па.
Ответ: Давление газа в колбе термоса составляет примерно 1,24 Па.
Другие задания:
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