Question

In an adiabatic change, the pressure and temperature of a monoatomic gas are related with relation $p\propto T^c$ , where $C$ is equal to

• A. $\frac{5}{4}$
• B. $\frac{5}{3}$
• C. $\frac{5}{2}$
• D. $\frac{3}{5}$

Answer 1

Answer: (C)

Key Idea Poisson's equation for adiabatic process is given by
$p{V}^{\gamma }$ =constant
For adiabatic process, Poisson’s equation is given by
$P{V}^{\gamma }$ = constant $\dots (i)$
Ideal gas relation is
$pV=RT$
$\Rightarrow V=\frac{RT}{p}\dots (ii)$
From Eqs. (i) and (ii), we get
$p{\left(\frac{RT}{p}\right)}^{\gamma }=$ constant
$\Rightarrow \frac{T^{\gamma }}{p^{\gamma -1}}=$ constant $\dots (iii)$
where $\gamma$ is ratio of specific heats of the gas
Given, $p\propto T^C$ $\dots (iv)$
On comparing with Eq. (iii), we have
$C=\frac{\gamma }{\gamma -1}$
For a monoatomic gas $\gamma =\frac{5}{3}$
We have
$C=\frac{\frac{5}{3}}{\frac{5}{3}-1}=\frac{5}{2}$