Q53 / 180 · COMEDK 2022 · Chemistry
The difference between heat capacity at constant pressure and heat capacity at constant volume is
- Correct answerA$R$
- B$\dfrac{R}{T}$
- C$\dfrac{R}{V}$
- D1
Solution
For an ideal gas, the enthalpy $H$ is defined as $H = U + PV$.
Differentiating with respect to temperature $T$ at constant pressure, we have $\left( \dfrac{\partial H}{\partial T} \right)_P = \left( \dfrac{\partial U}{\partial T} \right)_P + P \left( \dfrac{\partial V}{\partial T} \right)_P$.
By definition, $C_P = \left( \dfrac{\partial H}{\partial T} \right)_P$ and $C_V = \left( \dfrac{\partial U}{\partial T} \right)_V$. For an ideal gas, $U$ depends only on temperature, so $\left( \dfrac{\partial U}{\partial T} \right)_P = \left( \dfrac{\partial U}{\partial T} \right)_V = C_V$.
Using the ideal gas equation $PV = nRT$, for one mole ($n=1$), $PV = RT$. Differentiating with respect to $T$ at constant pressure gives $P \left( \dfrac{\partial V}{\partial T} \right)_P = R$.
Substituting these into the enthalpy derivative equation, we get $C_P = C_V + R$, which implies $C_P - C_V = R$.
Answer: $R$
Differentiating with respect to temperature $T$ at constant pressure, we have $\left( \dfrac{\partial H}{\partial T} \right)_P = \left( \dfrac{\partial U}{\partial T} \right)_P + P \left( \dfrac{\partial V}{\partial T} \right)_P$.
By definition, $C_P = \left( \dfrac{\partial H}{\partial T} \right)_P$ and $C_V = \left( \dfrac{\partial U}{\partial T} \right)_V$. For an ideal gas, $U$ depends only on temperature, so $\left( \dfrac{\partial U}{\partial T} \right)_P = \left( \dfrac{\partial U}{\partial T} \right)_V = C_V$.
Using the ideal gas equation $PV = nRT$, for one mole ($n=1$), $PV = RT$. Differentiating with respect to $T$ at constant pressure gives $P \left( \dfrac{\partial V}{\partial T} \right)_P = R$.
Substituting these into the enthalpy derivative equation, we get $C_P = C_V + R$, which implies $C_P - C_V = R$.
Answer: $R$
Hence, the correct answer is option: 'A'.