Difference between revisions of "Ideal gas: Heat capacity"

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m (Further re-write.)
m (Minor touches.)
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:<math>C_V = \left. \frac{\partial U}{\partial T} \right\vert_V </math>
 
:<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  
+
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>
+
:<math>C_V =  \frac{\partial ~ }{\partial T}  \left( \frac{3}{2}RT \right) = \frac{3}{2} R </math>.
  
 
At constant [[pressure]] one has
 
At constant [[pressure]] one has
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thus  
 
thus  
  
:<math>C_p =  C_v + R </math>
+
:<math>C_p =  C_v + R  =  \frac{5}{2} R</math>
  
 
where <math>R</math> is the [[molar gas constant]].
 
where <math>R</math> is the [[molar gas constant]].

Revision as of 17:46, 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 .

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

  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