Difference between revisions of "Ideal gas: Heat capacity"

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:<math>C_p - C_V = \left.\frac{\partial V}{\partial T}\right\vert_p \left(p + \left.\frac{\partial E}{\partial V}\right\vert_T \right) </math>
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The [[heat capacity]] at constant volume is given by
:<math>\left.C_p -C_V \right.=R</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]].
 
==References==
 
==References==
 
#Donald A. McQuarrie "Statistical Mechanics" (1976) Eq. 1-1
 
#Donald A. McQuarrie "Statistical Mechanics" (1976) Eq. 1-1
 
#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]]

Latest revision as of 16:34, 11 May 2012

The heat capacity at constant volume is given by

C_V = \left. \frac{\partial U}{\partial T} \right\vert_V

where U is the internal energy. Given that an ideal gas has no interatomic potential energy, the only term that is important is the kinetic energy of an ideal gas, which is equal to (3/2)RT. Thus

C_V =  \frac{\partial ~ }{\partial T}  \left( \frac{3}{2}RT \right) = \frac{3}{2} R

At constant pressure one has

C_p = \left. \frac{\partial U}{\partial T} \right\vert_p + p \left.\frac{\partial V}{\partial T} \right\vert_p

we can see that, just as before, one has

\left. \frac{\partial U}{\partial T} \right\vert_p = \frac{3}{2} R

and from the equation of state of an ideal gas

p \left.\frac{\partial V}{\partial T} \right\vert_p = \frac{\partial }{\partial T} (RT) = R

thus

C_p =  C_v + R  =  \frac{5}{2} R

where R is the molar gas constant.

References[edit]

  1. Donald A. McQuarrie "Statistical Mechanics" (1976) Eq. 1-1
  2. Landau and Lifshitz Course of Theoretical Physics Volume 5 Statistical Physics 3rd Edition Part 1 Equation 42.11