Question

During an adiabatic process, the pressure of a gas is found to be proportional to the cube of its absolute temperature. The ratio $\frac{C_P}{C_V}$ for the gas is

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

Answer 1

Answer: (A)

In an adiabatic process, ${\left(\text{T}\right)}^{\gamma }=\left(\text{constant}\right){\left(\text{P}\right)}^{\gamma -1}$
or ${\left(\text{T}\right)}^{\gamma /\gamma -1}=\left(\text{constant}\right)\text{ P}$
Given ${\left(\text{T}\right)}^3=\left(\text{constant}\right)\text{ P}$
$\therefore$ $\frac{\gamma }{\gamma -1}=3\Rightarrow 3\gamma -3=\gamma$
or $2\gamma =3\Rightarrow \gamma =3/2$
For monoatomic gas, $\gamma =\frac{5}{3}$ = 1.67
For diatomic gas, $\gamma =\frac{7}{5}$ = 1.4
when $\gamma$ = 1.5, the gas must be a suitable mixture of monoatomic and diatomic gases
$\therefore$ $\gamma$ = 3/2.