text: This is a derivation to obtain an expression for  for an ideal gas.
  An ideal gas has the equation of state:  
  where
  :P = pressure
  :V = volume
  :n = number of moles
  :R = universal gas constant
  :T = temperature
  The ideal gas equation of state can be arranged to give:
  :    or  
  The following partial derivatives are obtained from the above equation of state:
  The following simple expressions are obtained for thermal expansion coefficient :
  and for isothermal compressibility :
  One can now calculate  for ideal gases from the previously obtained general formula:
  Substituting from the ideal gas equation gives finally:
  where n = number of moles of gas in the thermodynamic system under consideration and R = universal gas constant.  On a per mole basis, the expression for difference in molar heat capacities becomes simply R for ideal gases as follows:
  This result would be consistent if the specific difference were derived directly from the general expression for .