{"id":231820,"date":"2025-06-11T12:07:34","date_gmt":"2025-06-11T12:07:34","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=231820"},"modified":"2025-06-11T12:07:36","modified_gmt":"2025-06-11T12:07:36","slug":"an-ideal-monatomic-gas-has-a-molar-heat-capacity-cmp-at-constant-pressure","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/11\/an-ideal-monatomic-gas-has-a-molar-heat-capacity-cmp-at-constant-pressure\/","title":{"rendered":"An ideal monatomic gas has a molar heat capacity Cmp at constant pressure."},"content":{"rendered":"\n

An ideal monatomic gas has a molar heat capacity Cmp at constant pressure. What is the molar heat capacity at constant volume of an ideal diatomic gas?<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Answer:<\/strong>
For an ideal diatomic gas, the molar heat capacity at constant volume CVC_VCV\u200b is approximately:CV=52RC_V = \\frac{5}{2} RCV\u200b=25\u200bR<\/p>\n\n\n\n

where RRR is the universal gas constant.<\/p>\n\n\n\n

\n\n\n\n

Explanation:<\/strong>
The heat capacity of a gas depends on the number of degrees of freedom available to its molecules. Degrees of freedom represent independent ways in which a molecule can store energy \u2014 through translation, rotation, and vibration.<\/p>\n\n\n\n

For an ideal monatomic<\/strong> gas, such as helium or argon, only translational motion contributes significantly to its internal energy. A monatomic gas molecule moves in three dimensions, giving it 3 translational degrees of freedom. According to the equipartition theorem, each degree of freedom contributes 12kT\\frac{1}{2}kT21\u200bkT of energy per molecule, where kkk is Boltzmann\u2019s constant and TTT is the temperature. On a molar basis, this gives:U=32nRTU = \\frac{3}{2}nRTU=23\u200bnRT<\/p>\n\n\n\n

Thus, the molar heat capacity at constant volume for a monatomic gas is:CV=(\u2202U\u2202T)V=32RC_V = \\left( \\frac{\\partial U}{\\partial T} \\right)_V = \\frac{3}{2}RCV\u200b=(\u2202T\u2202U\u200b)V\u200b=23\u200bR<\/p>\n\n\n\n

For an ideal diatomic<\/strong> gas (such as nitrogen or oxygen at moderate temperatures), both translational and rotational motions contribute. Diatomic molecules have 3 translational and 2 rotational degrees of freedom (rotation around the axis along the bond is negligible at room temperature due to quantum mechanical restrictions). Thus, they have a total of 5 degrees of freedom contributing to internal energy. The internal energy becomes:U=52nRTU = \\frac{5}{2}nRTU=25\u200bnRT<\/p>\n\n\n\n

Hence, the molar heat capacity at constant volume is:CV=(\u2202U\u2202T)V=52RC_V = \\left( \\frac{\\partial U}{\\partial T} \\right)_V = \\frac{5}{2}RCV\u200b=(\u2202T\u2202U\u200b)V\u200b=25\u200bR<\/p>\n\n\n\n

At higher temperatures, vibrational modes may become active, increasing CVC_VCV\u200b, but at ordinary temperatures, 52R\\frac{5}{2}R25\u200bR is the standard value used.<\/p>\n\n\n\n

\"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

An ideal monatomic gas has a molar heat capacity Cmp at constant pressure. What is the molar heat capacity at constant volume of an ideal diatomic gas? The Correct Answer and Explanation is: Answer:For an ideal diatomic gas, the molar heat capacity at constant volume CVC_VCV\u200b is approximately:CV=52RC_V = \\frac{5}{2} RCV\u200b=25\u200bR where RRR is the […]<\/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-231820","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/231820","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=231820"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/231820\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=231820"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=231820"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=231820"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}