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| The [[heat capacity]] at constant volume is given by
| | :<math>\left.c_p -c_v \right.=1</math> |
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| :<math>C_V = \left. \frac{\partial U}{\partial T} \right\vert_V </math>
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| where <math>U</math> is the [[internal energy]]. Given that an [[ideal gas]] has no interatomic potential energy, the only term that is important is the [[Ideal gas: Energy | kinetic energy of an ideal gas]], which is equal to <math>(3/2)RT</math>. Thus
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| :<math>C_V = \frac{\partial ~ }{\partial T} \left( \frac{3}{2}RT \right) = \frac{3}{2} R </math>
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| At constant [[pressure]] one has
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| :<math>C_p = \left. \frac{\partial U}{\partial T} \right\vert_p + p \left.\frac{\partial V}{\partial T} \right\vert_p</math>
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| we can see that, just as before, one has
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| :<math>\left. \frac{\partial U}{\partial T} \right\vert_p = \frac{3}{2} R </math> | |
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| and from the [[Equation of State: Ideal Gas | equation of state of an ideal gas]]
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| :<math>p \left.\frac{\partial V}{\partial T} \right\vert_p = \frac{\partial }{\partial T} (RT) = R</math>
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| thus
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| :<math>C_p = C_v + R = \frac{5}{2} R</math>
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| where <math>R</math> is the [[molar gas constant]].
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| ==References== | | ==References== |
| #Donald A. McQuarrie "Statistical Mechanics" (1976) Eq. 1-1
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| #Landau and Lifshitz Course of Theoretical Physics Volume 5 Statistical Physics 3rd Edition Part 1 Equation 42.11 | | #Landau and Lifshitz Course of Theoretical Physics Volume 5 Statistical Physics 3rd Edition Part 1 Equation 42.11 |
| [[Category: Ideal gas]] | | [[Category: Ideal gas]] |