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Q.
For an ideal gas, the molar specific heat capacities at constant pressure and volume are $C_{p}$ and $C_{V}$ respectively. Choose the correct option.
Thermodynamics
Solution:
Using first law of thermodynamics,
$\Delta Q=\Delta U+p \Delta V$
At constant $V, \Delta V=0$
$C_{V}=\left(\frac{\Delta Q}{\Delta T}\right)_{V}=\left(\frac{\Delta U}{\Delta T}\right)_{V}$
At constant pressure $p, \Delta p=0$
$C_{p}=\left(\frac{\Delta Q}{\Delta T}\right)_{p}=\left(\frac{\Delta U}{\Delta T}\right)_{p}+\left(\frac{p \Delta V}{\Delta T}\right)_{p}$
$=\left(\frac{\Delta U}{\Delta T}\right)_{p}+p\left(\frac{\Delta V}{\Delta T}\right)_{p}$
From ideal gas equation,
$p V=\mu R T$
For $\mu=1, p V=R T$
At constant pressure $p, $
$p \Delta V=R \Delta T$
$p\left(\frac{\Delta V}{\Delta T}\right)_{p}=R$
Thus, all the equations given in the options are correct for an ideal gas.