One mole of an ideal gas expands adiabatically from an initial state to final state . Another mole of the same gas expands isothermally from a different initial state to the same final state . The ratio of the specific heats at constant pressure and con

Question

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 $T_{A} / T_{B}$ ? [2023]

Smart Achievers · Accepted Answer

(1)

$For adiabatic process$ $T V^{γ - 1} = constant$

$∴$ $T_{A} V_{0}^{γ - 1} = T_{f} {(5 V_{0})}^{γ - 1} ...(i)$

and for isothermal process, no change in temperature

$∴$ $T_{B} = T_{f} ...(ii)$

Dividing (i) by (ii)

$\frac{T_{A}}{T_{B}} = 5^{γ - 1}$

Solution

1 $5^{γ - 1}$

2 $5^{1 - γ}$

3 $5^{γ}$

4 $5^{1 + γ}$

Ans.
(1)

$For adiabatic process$ $T V^{γ - 1} = constant$

$∴$ $T_{A} V_{0}^{γ - 1} = T_{f} {(5 V_{0})}^{γ - 1} ...(i)$

and for isothermal process, no change in temperature

$∴$ $T_{B} = T_{f} ...(ii)$

Dividing (i) by (ii)

$\frac{T_{A}}{T_{B}} = 5^{γ - 1}$

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Solution

1 5γ-1

2 51-γ

3 5γ

4 51+γ