Question

In an adiabatic change, the pressure and temperature of a monoatomic gas are related with relation as $P\propto T^C$, 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)

For adiabatic process, Poisson's equation is given by
$P{V}^{\gamma }=\text{ constant }$(1)
Ideal gas relation is
$\Rightarrow V=\frac{RT}{P}$(2)
From Eqs. (1) and (2), we get
$P{\left(\frac{RT}{P}\right)}^{\gamma }=\text{ constant }$
$\Rightarrow \frac{T^{\gamma }}{P^{\gamma -1}}=\text{ constant }$(3)
where $\gamma$ is ratio of specific heats of the gas.
Given, $P\propto T^C$ (4)
On comparing with Eq. (3), we have, $C=\frac{\gamma }{\gamma -1}$
For a monoatomic gas $\gamma =\frac{5}{3}$
$\therefore$ We have, $C=\frac{\frac{5}{3}}{\frac{5}{3}-1}=\frac{5}{2}$