A one mole of an ideal gas expands adiabatically at constant pressure such that its temperature T propto (1/√ V) . The value of the adiabatic constant of gas is

Question

A one mole of an ideal gas expands adiabatically at constant pressure such that its temperature T propto (1/√ V) . The value of the adiabatic constant of gas is (A)  1.3  (B)  1.5  (C)  1.67  (D)  2.0

Tardigrade Admin · Accepted Answer

In an adiabatic expansion, PV&gamma; = constant ∵ P = (RT/V) (for 1 mole of gas ) ∴ ((RT/V))V&gamma; = constant TV&gamma; -1 = constant or T propto V1-&gamma; As T propto (1/√V) ∴ 1- &gamma; = - (1/2) &gamma; = 1 +(1/2) = (3/2) = 1.5

Q. A one mole of an ideal gas expands adiabatically at constant pressure such that its temperature $T \propto \frac{1}{V}$ . The value of the adiabatic constant of gas is

$1.3$

$1.5$

$1.67$

$2.0$

In an adiabatic expansion,
$P V^{γ} =$ constant
$∵ P = \frac{RT}{V}$ (for $1$ mole of gas )
$∴ (\frac{RT}{V}) V^{γ} =$ constant
$T V^{γ - 1} =$ constant or $T \propto V^{1 - γ}$
As $T \propto \frac{1}{V}$
$∴ 1 - γ = - \frac{1}{2}$
$γ = 1 + \frac{1}{2} = \frac{3}{2} = 1.5$

Q. A one mole of an ideal gas expands adiabatically at constant pressure such that its temperature T∝V​1​ . The value of the adiabatic constant of gas is

1.3

1.5

1.67

2.0