{"id":194043,"date":"2025-02-22T07:27:42","date_gmt":"2025-02-22T07:27:42","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=194043"},"modified":"2025-02-22T07:27:44","modified_gmt":"2025-02-22T07:27:44","slug":"for-aluminum-the-heat-capacity-at-constant-volume-cv-at-30-k-is-0-81-j-mol-k-and-the-debye-temperature-is-375-k","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/22\/for-aluminum-the-heat-capacity-at-constant-volume-cv-at-30-k-is-0-81-j-mol-k-and-the-debye-temperature-is-375-k\/","title":{"rendered":"For aluminum, the heat capacity at constant volume Cv at 30 K is 0.81 J\/mol-K, and the Debye temperature is 375 K"},"content":{"rendered":"\n<p>For aluminum, the heat capacity at constant volume Cv at 30 K is 0.81 J\/mol-K, and the Debye temperature is 375 K. Estimate the specific heat<br>(a) At 50 K and<br>(b) At 425 K.<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>To estimate the specific heat of aluminum at 50 K and 425 K, we use the <strong>Debye model<\/strong>, which describes the heat capacity of solids at different temperature ranges.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Debye Model Formula for Heat Capacity:<\/strong><\/h3>\n\n\n\n<p>The Debye model gives the heat capacity at constant volume as:<\/p>\n\n\n\n<p>[<br>C_V = 9Nk_B \\left(\\frac{T}{\\Theta_D}\\right)^3 \\int_0^{\\Theta_D\/T} \\frac{x^4 e^x}{(e^x -1)^2} dx<br>]<\/p>\n\n\n\n<p>where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( C_V ) = heat capacity at constant volume<\/li>\n\n\n\n<li>( N ) = number of atoms per mole (Avogadro\u2019s number)<\/li>\n\n\n\n<li>( k_B ) = Boltzmann constant<\/li>\n\n\n\n<li>( T ) = temperature<\/li>\n\n\n\n<li>( \\Theta_D ) = Debye temperature<\/li>\n\n\n\n<li>( x = \\frac{h \\nu}{k_B T} ), integration variable<\/li>\n<\/ul>\n\n\n\n<p>However, in practical calculations, we use approximations based on temperature ranges:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>For ( T \\ll \\Theta_D ) (low-temperature regime, ( T &lt; 0.1\\Theta_D )):<\/strong><br>[<br>C_V \\approx \\frac{12\\pi^4}{5} Nk_B \\left(\\frac{T}{\\Theta_D}\\right)^3<br>]<br>This follows the <strong>Debye (T^3) Law<\/strong>.<\/li>\n\n\n\n<li><strong>For ( T \\gg \\Theta_D ) (high-temperature regime, ( T > \\Theta_D )):<\/strong><br>[<br>C_V \\approx 3Nk_B<br>]<br>This approaches the <strong>Dulong\u2013Petit limit<\/strong> (~25 J\/mol-K for aluminum).<\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>(a) Specific Heat at 50 K<\/strong><\/h3>\n\n\n\n<p>Since ( 50 K &lt; \\Theta_D ), we use the Debye approximation:<br>[<br>C_V(50K) = C_V(30K) \\times \\left(\\frac{50}{30}\\right)^3<br>]<\/p>\n\n\n\n<p>[<br>C_V(50K) = 0.81 \\times \\left(\\frac{50}{30}\\right)^3<br>]<\/p>\n\n\n\n<p>[<br>C_V(50K) \\approx 3.74 J\/mol-K<br>]<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>(b) Specific Heat at 425 K<\/strong><\/h3>\n\n\n\n<p>Since ( 425 K &gt; \\Theta_D ), aluminum is in the classical regime, where the heat capacity approaches the <strong>Dulong\u2013Petit law<\/strong>:<\/p>\n\n\n\n<p>[<br>C_V(425K) \\approx 3R = 24.9 J\/mol-K<br>]<\/p>\n\n\n\n<p>where ( R ) is the gas constant (( 8.314 J\/mol-K )).<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Final Answers:<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>( C_V(50K) \\approx 3.74 J\/mol-K )<\/strong><\/li>\n\n\n\n<li><strong>( C_V(425K) \\approx 24.9 J\/mol-K )<\/strong> (approaching the classical limit)<\/li>\n<\/ul>\n\n\n\n<p>I&#8217;ll now generate an image to illustrate this.<\/p>\n\n\n\n<p>Here is the generated diagram illustrating the heat capacity of aluminum as a function of temperature. It shows the two main regions: the low-temperature Debye ( T^3 ) regime and the high-temperature Dulong-Petit limit. Let me know if you need any modifications!<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/02\/image-1208.png\" alt=\"\" class=\"wp-image-194044\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>For aluminum, the heat capacity at constant volume Cv at 30 K is 0.81 J\/mol-K, and the Debye temperature is 375 K. Estimate the specific heat(a) At 50 K and(b) At 425 K. The Correct Answer and Explanation is : To estimate the specific heat of aluminum at 50 K and 425 K, we use [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-194043","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/194043","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=194043"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/194043\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=194043"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=194043"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=194043"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}