First slide

Different type of processes

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

Moderate

A

$\frac{5}{4}$
B

$\frac{5}{3}$
C

$\frac{3}{5}$
D

$\frac{5}{2}$

Solution

For adiabatic process,
${PV}^{γ}$ = constant----------------(i)
We have ideal gas relation, PV = RT
$\Rightarrow V = \frac{R T}{P}$ -----------------(ii)
From Eqs. (i) and (ii), we get
$P {(\frac{RT}{P})}^{γ} = constant$
$\Rightarrow \frac{T^{γ}}{P^{γ - 1}} = constant$ -------------(iii)
where $γ$ is adiabatic constant of the gas.
Given, $P \propto T^{C}$
On comparing with Eq. (iii), we have,
$C = \frac{γ}{γ - 1}$
For a monoatomic gas $γ = \frac{5}{3}$
$∴ We have, C = \frac{\frac{5}{3}}{\frac{5}{3} - 1} = \frac{5}{2}$

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