We have defined specific heat capacity and molar specific heat capacity earlier in the previous chapter.
There are two specific heats of ideal gases.
(i) Specific heat capacity at constant volume
(ii) Specific heat capacity at constant pressure
Cp and Cv are molar specific heat capacities of ideal gas at constant pressure and volume respectively for Cp and Cv of ideal gas there is a simple relation.
$C_p-C_v = R$ ---(1)
where R- universal gas constant
This relation can be proved as follows from first law of thermodynamics for 1 mole of gas we have
$ \Delta Q =\Delta U+P \Delta V $ --(2)
If heat is absorbed at constant volume for the temperature difference $\Delta T$ then ΔV = 0 and
CV=(ΔQ/ΔT)V=(ΔU/ΔT)V -----(3)
If Q in absorbed at constant pressure for the temperature difference $\Delta T$,then
CP=(ΔQ/ΔT)P=(ΔU/ΔT)P+P(ΔV/ΔT)P ---- (4)
Now ideal gas equation for 1 mole of gas is
PV = RT
P(ΔV/ΔT) = R --(5)
From (3) and (4)
CP - CV=(ΔU/ΔT)P-(ΔU/ΔT)V+P(ΔV/ΔT)P
Now from equation (5)
CP - CV=(ΔU/ΔT)P-(ΔU/ΔT)V + R
Since internal energy U of ideal gas depends only on temperature so subscripts P and V have no meaning.
$C_P - C_V = R$
which is the desired relation
Relation Between CV and Internal Energy
From equation (3)
CV=(ΔQ/ΔT)V=(ΔU/ΔT)V
or
$\Delta U = C_v \Delta T$
or
$dU= C_v dT$
for n moles for gas
$dU= nC_v dT$
If we take the internal energy at T=0,then
$U=nC_v T$
