# Difference between revisions of "Ideal gas: Heat capacity"

Carl McBride (talk | contribs) m (Slight re-write.) |
Carl McBride (talk | contribs) m (Further re-write.) |
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:<math>C_V = \frac{\partial ~ }{\partial T} \left( \frac{3}{2}RT \right) = \frac{3}{2} R </math> | :<math>C_V = \frac{\partial ~ }{\partial T} \left( \frac{3}{2}RT \right) = \frac{3}{2} R </math> | ||

+ | At constant [[pressure]] one has | ||

− | + | :<math>C_p = \left. \frac{\partial U}{\partial T} \right\vert_p + p \left.\frac{\partial V}{\partial T} \right\vert_p</math> | |

− | + | we can see that, just as before, one has | |

− | + | :<math>\left. \frac{\partial U}{\partial T} \right\vert_p = \frac{3}{2} R </math> | |

+ | |||

+ | and from the [[Equation of State: Ideal Gas | equation of state of an ideal gas]] | ||

+ | |||

+ | :<math>p \left.\frac{\partial V}{\partial T} \right\vert_p = \frac{\partial }{\partial T} (RT) = R</math> | ||

+ | |||

+ | thus | ||

+ | |||

+ | :<math>C_p = C_v + R </math> | ||

− | |||

where <math>R</math> is the [[molar gas constant]]. | where <math>R</math> is the [[molar gas constant]]. | ||

==References== | ==References== |

## Revision as of 17:43, 4 December 2008

The heat capacity at constant volume is given by

where 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 . Thus

At constant pressure one has

we can see that, just as before, one has

and from the equation of state of an ideal gas

thus

where is the molar gas constant.

## References

- 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