{"id":35540,"date":"2025-06-24T12:04:11","date_gmt":"2025-06-24T12:04:11","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=35540"},"modified":"2025-06-24T12:04:13","modified_gmt":"2025-06-24T12:04:13","slug":"a-football-at-27c-has-0-5-mole-of-air-molecules","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/a-football-at-27c-has-0-5-mole-of-air-molecules\/","title":{"rendered":"A football at 27\u00b0C has 0.5\u2006mole of air molecules"},"content":{"rendered":"\n

A football at 27\u00b0C has 0.5\u2006mole of air molecules. Calculate the internal energy of air in the ball<\/p>\n\n\n\n

The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Correct Answer:<\/strong><\/p>\n\n\n\n

The internal energy (U) of air in the football is 1,868.5 J (joules)<\/strong>.<\/p>\n\n\n\n

\n\n\n\n

Explanation:<\/strong><\/p>\n\n\n\n

To calculate the internal energy of air in the football, we assume the air behaves like an ideal diatomic gas<\/strong>, which is appropriate because air is mostly nitrogen (N\u2082) and oxygen (O\u2082), both diatomic gases at room temperature.<\/p>\n\n\n\n

Step 1: Use the internal energy formula for an ideal diatomic gas<\/h3>\n\n\n\n

U=52nRTU = \\frac{5}{2} nRT<\/p>\n\n\n\n

Where:<\/p>\n\n\n\n

    \n
  • UU = internal energy (J)<\/li>\n\n\n\n
  • nn = number of moles = 0.5<\/li>\n\n\n\n
  • RR = ideal gas constant = 8.314 J\/mol\u00b7K<\/li>\n\n\n\n
  • TT = temperature in Kelvin<\/li>\n<\/ul>\n\n\n\n

    Step 2: Convert Celsius to Kelvin<\/h3>\n\n\n\n

    T=27\u00b0C+273.15=300.15 KT = 27\u00b0C + 273.15 = 300.15 \\text{ K}<\/p>\n\n\n\n

    Step 3: Plug values into the formula<\/h3>\n\n\n\n

    U=52\u00d70.5\u00d78.314\u00d7300.15U = \\frac{5}{2} \\times 0.5 \\times 8.314 \\times 300.15 U=1.25\u00d78.314\u00d7300.15U = 1.25 \\times 8.314 \\times 300.15 U\u22481.25\u00d72492.3U \u2248 1.25 \\times 2492.3 U\u22483115.4 JU \u2248 3115.4 \\text{ J}<\/p>\n\n\n\n

    Correction:<\/strong> Let\u2019s compute more precisely: U=52\u00d70.5\u00d78.314\u00d7300.15=2.5\u00d70.5\u00d78.314\u00d7300.15=1.25\u00d78.314\u00d7300.15\u22481.25\u00d72491.41\u22483,114.26 JU = \\frac{5}{2} \\times 0.5 \\times 8.314 \\times 300.15 = 2.5 \\times 0.5 \\times 8.314 \\times 300.15 = 1.25 \\times 8.314 \\times 300.15 \u2248 1.25 \\times 2491.41 \u2248 3,114.26 \\text{ J}<\/p>\n\n\n\n

    Actually, the correct final value is 3,114.26 J<\/strong>.<\/p>\n\n\n\n

    There was an earlier error in the initial number reported. Therefore, the correct internal energy<\/strong> is approximately: 3,114.26 J\\boxed{3,114.26 \\text{ J}}<\/p>\n\n\n\n

    \n\n\n\n

    Why this formula is used:<\/h3>\n\n\n\n

    In kinetic theory, diatomic gases at room temperature have 5 degrees of freedom<\/strong>: 3 translational and 2 rotational. Vibrational modes are not fully active at 27\u00b0C, so the internal energy per mole is: 52RT\\frac{5}{2}RT<\/p>\n\n\n\n

    This internal energy accounts for the kinetic energy stored in the motion of the molecules. Since the air molecules are moving and colliding inside the ball, they store energy that depends only on the temperature and the number of molecules.<\/p>\n","protected":false},"excerpt":{"rendered":"

    A football at 27\u00b0C has 0.5\u2006mole of air molecules. Calculate the internal energy of air in the ball The correct answer and explanation is: Correct Answer: The internal energy (U) of air in the football is 1,868.5 J (joules). Explanation: To calculate the internal energy of air in the football, we assume the air behaves […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-35540","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/35540","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=35540"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/35540\/revisions"}],"predecessor-version":[{"id":35542,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/35540\/revisions\/35542"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=35540"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=35540"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=35540"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}