Show that the most probable distribution of 2N molecules of an idea...

Question
Show that the most probable distribution of 2N molecules of an ideal gas contained in two equal and connected volumes at the same temperature is N molecules in each volume.
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      • eloquent.retrieved: App\Models\SubjectCat (103μs)
      • Illuminate\Cache\Events\KeyWritten (1.43ms)
      • Illuminate\Cache\Events\CacheHit (307μs)
      • Illuminate\Cache\Events\CacheHit (326μs)
      • Illuminate\Cache\Events\CacheHit (429μs)
      • Illuminate\Cache\Events\CacheHit (478μs)
      • Illuminate\Cache\Events\CacheHit (803μs)
      • Illuminate\Cache\Events\CacheHit (352μs)
      • Illuminate\Cache\Events\CacheHit (184μs)
      • Illuminate\Cache\Events\CacheHit (276μs)
      • Illuminate\Cache\Events\CacheHit (491μs)
      • Illuminate\Cache\Events\CacheHit (494μs)
      • Illuminate\Cache\Events\CacheHit (205μs)
      • Illuminate\Cache\Events\CacheHit (200μs)
      • Illuminate\Cache\Events\CacheHit (225μs)
      • Illuminate\Cache\Events\CacheHit (201μs)
      • Illuminate\Cache\Events\CacheHit (168μs)
      • Illuminate\Cache\Events\CacheHit (189μs)
      • Illuminate\Cache\Events\CacheHit (165μs)
      • creating: site.layouts.app (518μs)
      • composing: site.layouts.app (25μs)
      • creating: components.canonical (547μs)
      • composing: components.canonical (105μs)
      • creating: components.open-graph (396μs)
      • composing: components.open-graph (88μs)
      • creating: site.headers.header (446μs)
      • composing: site.headers.header (84μs)
      • Illuminate\Cache\Events\CacheHit (2.55ms)
      • creating: components.footer (154μs)
      • composing: components.footer (155μs)
      • Illuminate\Cache\Events\CacheHit (1.1ms)
      • Illuminate\Cache\Events\CacheMissed (705μs)
      • Illuminate\Database\Events\StatementPrepared (1.49ms)
      • Illuminate\Database\Events\QueryExecuted (1.43ms)
      • eloquent.retrieved: App\Models\SubjectCat (108μs)
      • Illuminate\Cache\Events\KeyWritten (603μs)
      • Illuminate\Cache\Events\CacheHit (341μs)
      • Illuminate\Cache\Events\CacheHit (147μs)
      • Illuminate\Cache\Events\CacheHit (125μs)
      • Illuminate\Cache\Events\CacheHit (115μs)
      • Illuminate\Cache\Events\CacheHit (116μs)
      • Illuminate\Cache\Events\CacheHit (358μs)
      • Illuminate\Cache\Events\CacheHit (336μs)
      • Illuminate\Cache\Events\CacheHit (211μs)
      • Illuminate\Cache\Events\CacheHit (188μs)
      • Illuminate\Cache\Events\CacheHit (190μs)
      • Illuminate\Cache\Events\CacheHit (218μs)
      • Illuminate\Cache\Events\CacheHit (310μs)
      • creating: components.forms.contact-us (285μs)
      • composing: components.forms.contact-us (194μs)
      • creating: components.forms.get-started (344μs)
      • composing: components.forms.get-started (174μs)
      • creating: components.forms.free-tool-download (213μs)
      • composing: components.forms.free-tool-download (90μs)
      • creating: components.forms.claim-free-worksheet (152μs)
      • composing: components.forms.claim-free-worksheet (90μs)
      • creating: components.forms.tutor-subscription-waitlist (243μs)
      • composing: components.forms.tutor-subscription-waitlist (85μs)
      • creating: components.forms.tutor-subscription-join (146μs)
      • composing: components.forms.tutor-subscription-join (72μs)
      • creating: components.forms.tutor-support (125μs)
      • composing: components.forms.tutor-support (69μs)
      • 315 x Illuminate\Cache\Events\CacheHit (19.79%)
        130ms
        16 x Illuminate\Database\Events\StatementPrepared (4.9%)
        32.24ms
        16 x Illuminate\Database\Events\QueryExecuted (4.59%)
        30.15ms
        7 x Illuminate\Cache\Events\KeyWritten (2.02%)
        13.30ms
        1 x Illuminate\Foundation\Events\LocaleUpdated (0.86%)
        5.68ms
        7 x Illuminate\Cache\Events\CacheMissed (0.79%)
        5.20ms
        1 x Illuminate\Database\Events\ConnectionEstablished (0.44%)
        2.92ms
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (0.31%)
        2.04ms
        1 x Illuminate\Routing\Events\Routing (0.21%)
        1.36ms
        27 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.15%)
        974μs
        8 x eloquent.retrieved: App\Models\SubjectCat (0.11%)
        746μs
        1 x eloquent.booting: App\Models\SubjectCat (0.09%)
        603μs
        1 x creating: components.breadcrumbs (0.09%)
        577μs
        1 x creating: components.canonical (0.08%)
        547μs
        1 x eloquent.booting: App\Models\Subject (0.08%)
        539μs
        1 x creating: site.layouts.app (0.08%)
        518μs
        1 x creating: site.headers.header (0.07%)
        446μs
        1 x Illuminate\Routing\Events\RouteMatched (0.07%)
        445μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (0.06%)
        401μs
        1 x creating: components.open-graph (0.06%)
        396μs
        1 x creating: components.forms.get-started (0.05%)
        344μs
        7 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (0.05%)
        318μs
        1 x creating: homework.show (0.04%)
        292μs
        1 x creating: components.forms.contact-us (0.04%)
        285μs
        1 x creating: components.forms.tutor-subscription-waitlist (0.04%)
        243μs
        2 x eloquent.retrieved: App\Models\Subject (0.04%)
        242μs
        1 x creating: components.forms.free-tool-download (0.03%)
        213μs
        1 x composing: components.forms.contact-us (0.03%)
        194μs
        1 x composing: components.forms.get-started (0.03%)
        174μs
        1 x composing: components.footer (0.02%)
        155μs
        1 x creating: components.footer (0.02%)
        154μs
        1 x creating: components.forms.claim-free-worksheet (0.02%)
        152μs
        1 x creating: components.forms.tutor-subscription-join (0.02%)
        146μs
        1 x composing: components.breadcrumbs (0.02%)
        144μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.02%)
        125μs
        1 x creating: components.forms.tutor-support (0.02%)
        125μs
        1 x composing: homework.show (0.02%)
        123μs
        1 x composing: components.canonical (0.02%)
        105μs
        1 x eloquent.booted: App\Models\SubjectCat (0.01%)
        94μs
        1 x composing: components.forms.free-tool-download (0.01%)
        90μs
        1 x composing: components.forms.claim-free-worksheet (0.01%)
        90μs
        1 x composing: components.open-graph (0.01%)
        88μs
        1 x composing: components.forms.tutor-subscription-waitlist (0.01%)
        85μs
        1 x composing: site.headers.header (0.01%)
        84μs
        1 x eloquent.booted: App\Models\Subject (0.01%)
        72μs
        1 x composing: components.forms.tutor-subscription-join (0.01%)
        72μs
        1 x composing: components.forms.tutor-support (0.01%)
        69μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.01%)
        43μs
        1 x composing: site.layouts.app (0%)
        25μs
      14 templates were rendered
      • 1x homework.showshow.blade.phpblade
      • 1x components.breadcrumbsbreadcrumbs.blade.phpblade
      • 1x site.layouts.appapp.blade.phpblade
      • 1x components.canonicalcanonical.blade.phpblade
      • 1x components.open-graphopen-graph.blade.phpblade
      • 1x site.headers.headerheader.blade.phpblade
      • 1x components.footerfooter.blade.phpblade
      • 1x components.forms.contact-uscontact-us.blade.phpblade
      • 1x components.forms.get-startedget-started.blade.phpblade
      • 1x components.forms.free-tool-downloadfree-tool-download.blade.phpblade
      • 1x components.forms.claim-free-worksheetclaim-free-worksheet.blade.phpblade
      • 1x components.forms.tutor-subscription-waitlisttutor-subscription-waitlist.blade.phpblade
      • 1x components.forms.tutor-subscription-jointutor-subscription-join.blade.phpblade
      • 1x components.forms.tutor-supporttutor-support.blade.phpblade
      uri
      GET college-homework-library/{category}/{subject}/{id}
      middleware
      web, utm.parameters
      controller
      App\Http\Controllers\HomeworkLibraryController@show
      namespace
      where
      as
      homework.show
      file
      app/Http/Controllers/HomeworkLibraryController.php:79-176
      16 statements were executed, 5 of which were duplicates, 11 unique. Show only duplicated32.92ms
      • Connection Establishedtwenty4_siteHomeworkLibraryController.php#91
        Backtrace
        • 13. app/Http/Controllers/HomeworkLibraryController.php:91
        • 14. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 15. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 16. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary` where `status` = 'published' and `price` > 0 and `solutionslibrary`.`id` = '23016' limit 1
        7.17mstwenty4_siteHomeworkLibraryController.php#97
        Bindings
        • 0: published
        • 1: 0
        • 2: 23016
        Backtrace
        • 16. app/Http/Controllers/HomeworkLibraryController.php:97
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 18. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 19. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 20. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subjects` where `subjects`.`id` in (269)
        1.77mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` in (23016)
        940μstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subject_cats` where `subject_cats`.`id` = 34 limit 1
        1.58mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 34
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 26. app/Http/Controllers/HomeworkLibraryController.php:105
        • 27. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 28. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 29. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
      • select * from `solutionslibrary` where `id` <> 23016 and `subject` = 269 and `status` = 'published' and `price` > 0 order by RAND() limit 6
        3.79mstwenty4_siteHomeworkLibraryRepository.php#30
        Bindings
        • 0: 23016
        • 1: 269
        • 2: published
        • 3: 0
        Backtrace
        • 14. app/Repositories/HomeworkLibraryRepository.php:30
        • 15. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 16. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 18. app/Repositories/HomeworkLibraryRepository.php:39
        • 19. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `subjects` where `subjects`.`id` in (269)
        1.17mstwenty4_siteHomeworkLibraryRepository.php#30
        Backtrace
        • 19. app/Repositories/HomeworkLibraryRepository.php:30
        • 20. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 21. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 23. app/Repositories/HomeworkLibraryRepository.php:39
        • 24. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 23016 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'question' order by `order` asc, `id` asc
        2.39mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 23016
        • 1: question
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:58
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 23016 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'teaser' order by `order` asc, `id` asc
        2.73mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 23016
        • 1: teaser
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:69
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 23016 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'solution' order by `order` asc, `id` asc
        1.87mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 23016
        • 1: solution
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:80
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `subject_cats` where `subject_cats`.`id` = 34 limit 1
        1.4mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 34
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 32. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 33. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 34. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69
        • 35. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
      • select * from `subject_cats` where `subject_cats`.`id` = 33 limit 1
        1.03mstwenty4_siteSubject.php#100
        Bindings
        • 0: 33
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 11 limit 1
        1.24mstwenty4_siteSubject.php#100
        Bindings
        • 0: 11
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 5 limit 1
        1.1mstwenty4_siteSubject.php#100
        Bindings
        • 0: 5
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 34 limit 1
        1.43mstwenty4_siteSubject.php#100
        Bindings
        • 0: 34
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 31 limit 1
        1.44mstwenty4_siteSubject.php#100
        Bindings
        • 0: 31
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 36 limit 1
        1.87mstwenty4_siteSubject.php#100
        Bindings
        • 0: 36
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 28. view::components.footer:170
      App\Models\HomeworkLibrary\HomeworkLibraryFile
      27HomeworkLibraryFile.php
      App\Models\SubjectCat
      8SubjectCat.php
      App\Models\HomeworkLibrary\HomeworkLibrary
      7HomeworkLibrary.php
      App\Models\Subject
      2Subject.php
          _token
          9UJat3E7wpF4wUwPkpJ6ecSC7PgrKDJD2NoenxKn
          utm_source
          direct
          redirectUrl
          /college-homework-library/Physics/Thermodynamics/23016
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          array:1 [ "url" => "https://staging.dev.24houranswers.com/college-homework-library/Physics/Thermod...
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          PHPDEBUGBAR_STACK_DATA
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          status_code
          200
          
          status_text
          OK
          format
          html
          content_type
          text/html; charset=UTF-8
          request_query
          []
          
          request_request
          []
          
          request_headers
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          request_cookies
          []
          
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