Difference between revisions of "Ideal gas: Heat capacity"

From SklogWiki
Jump to: navigation, search
m (Slight re-write.)
Line 1: Line 1:
 +
The [[heat capacity]] at constant volume is given by
 +
 +
:<math>C_V = \left. \frac{\partial U}{\partial T} \right\vert_V </math>
 +
 +
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
 +
 +
:<math>C_V =  \frac{\partial ~ }{\partial T}  \left( \frac{3}{2}RT \right) = \frac{3}{2} R </math>
 +
 +
 +
One has
 +
 
:<math>C_p - C_V = \left.\frac{\partial V}{\partial T}\right\vert_p \left(p + \left.\frac{\partial U}{\partial V}\right\vert_T \right) </math>
 
:<math>C_p - C_V = \left.\frac{\partial V}{\partial T}\right\vert_p \left(p + \left.\frac{\partial U}{\partial V}\right\vert_T \right) </math>
  
Line 4: Line 15:
  
 
:<math>\left.C_p -C_V \right.=R</math>
 
:<math>\left.C_p -C_V \right.=R</math>
 +
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]]

Revision as of 15:41, 4 December 2008

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


One has

C_p - C_V = \left.\frac{\partial V}{\partial T}\right\vert_p \left(p + \left.\frac{\partial U}{\partial V}\right\vert_T \right)

for an ideal gas this becomes:

\left.C_p -C_V \right.=R

where R is the molar gas constant.

References

  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