One mole of an ideal gas expands adiabatically from an initial state ($T_{\mathrm{A}}, V_{0}$) to final state ($T_{\mathrm{f}}, 5 V_{0}$). Another mole of the same gas expands isothermally from a different initial state ($T_{\mathrm{B}}, \mathrm{V}_{0}$) to the same final state ($T_{\mathrm{f}}, 5 V_{0}$). The ratio of the specific heats at constant pressure and constant volume of this ideal gas is $\gamma$. What is the ratio $T_{\mathrm{A}} / T_{\mathrm{B}}$?

JEE Advanced · 2023 · 04 Jun 2023 · Paper 1 · Heat & Thermodynamics

One mole of an ideal gas expands adiabatically from an initial state (

T_{A}, V_{0}

) to final state (

T_{f}, 5 V_{0}

). Another mole of the same gas expands isothermally from a different initial state (

T_{B}, V_{0}

) to the same final state (

T_{f}, 5 V_{0}

). The ratio of the specific heats at constant pressure and constant volume of this ideal gas is

γ

. What is the ratio

T_{A} / T_{B}

Answer: (A)

For the adiabatic process:

T_{A} V_{0}^{γ - 1} = T_{f} (5 V_{0})^{γ - 1}

This gives:

\frac{T _{A}}{T _{f}} = 5^{γ - 1}

... (1)

For the isothermal process: Since the final temperature is

T_{f}

, and isothermal means constant temperature, we have

T_{B} = T_{f}

... (2)

From (1) and (2):

\frac{T _{A}}{T _{B}} = \frac{T _{A}}{T _{f}} = 5^{γ - 1}

JEE Advanced · 2023 · 04 Jun 2023 · Paper 1 · Single correct option