From the kinetic theory of ideal gases and Sackur-Tetrode equation, we derived the heat capacity for an ideal gas at constant volume to be CV : Nliig : 371R, where N is the number of molecules (and n is the number of moles). That is, adding heat (,2 to the system produces a temperature rise Q/CV. In this case all the input heat Q goes into raising the internal heat energy of the gas. Consider instead the case where we keep the pres-sure constant, but allow the volume to change according to PV : uRT. This gives us the heat capacity at constant pressure C12. We might accomplish this by having a movable piston which is moved as necessary to maintain a constant pressure P. Show that, for input of a small amount of heat Q, a the work done by the expanding gas on the piston is PAV, where AV is the change in volume; a the volume change AV is related to the temperature change of the gas via AV 2 'nRAT/P; a the change in the internal energy of the gas is Q PAV : Q -n..RAT; I- for n : 1 mole of gas, the ratio (7p 2 Q/AT : CV + R, which for an ideal gas gives 01) : 2 The value CI: is always larger than CV, because some amount of the input heat goes into the work of pushing the piston; that energy does not raise the temperature of the gas