Prove the relationship between $C_p$ and $C_v$ for an ideal gas.

(N/A) At constant volume,the heat capacity is denoted by $C_v$ and at constant pressure,it is denoted by $C_p$.
For a process at constant volume,the heat exchanged is $q_v = \Delta U = C_v \Delta T$.
For a process at constant pressure,the heat exchanged is $q_p = \Delta H = C_p \Delta T$.
For $1 \text{ mole}$ of an ideal gas,the enthalpy change is given by $\Delta H = \Delta U + \Delta (pV)$.
Since $pV = RT$ for $1 \text{ mole}$ of an ideal gas,we have $\Delta H = \Delta U + \Delta (RT)$.
Assuming $R$ is constant,$\Delta H = \Delta U + R \Delta T$.
Substituting the expressions for $\Delta H$ and $\Delta U$ in terms of heat capacities:
$C_p \Delta T = C_v \Delta T + R \Delta T$.
Dividing both sides by $\Delta T$,we get $C_p = C_v + R$ or $C_p - C_v = R$.