Question

During a reversible 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. $\frac{7}{2}$
• C. $\frac{5}{3}$
• D. $\frac{9}{7}$

Answer 1

Answer: (A)

For a reversible adiabatic process,

$p T^{\frac{\gamma}{1-\gamma}}=$ constant ....(i)

According to question, during a reversible adiabatic process the pressure of a gas is found to be proportional to the cube of its absolute temperature.

$p \propto T^{3} $

$p T^{-3}=$ constant .... (ii)

Equating equation (i) and (ii), we get

$\frac{\gamma}{1-\gamma} =-3 $

$\gamma =-3(1-\gamma) $

$\gamma =-3+3 \gamma$

$-2 \gamma =-3$

$\gamma =\frac{3}{2}$

As we know, the ratio of molar heat capacities at constant pressure and constant volume is respectively by $\gamma$. So, the ratio of $\frac{C_p}{C_v}$ for the gas is $\frac{3}{2}$.