Question

A given quantity of an ideal gas at pressure P and absolute temperature T obeys $P\alpha T^3$ during adiabatic process. The adiabatic bulk modulus of the gas is

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

Answer 1

The correct option is C $\frac{3}{2}P$

$\begin{array}{c}p\alpha T^3\\ p\alpha {\left(\frac{pv}{nR}\right)}^3\\ \Rightarrow p\alpha p^3{v}^3\\ \therefore p^2{v}^3=c\\ \Rightarrow p{v}^{\frac{3}{2}}=c\\ \Rightarrow p.\left(\frac{3}{2}{v}^{\frac{1}{2}}\right)+v^{\frac{3}{2}}.\frac{dp}{dv}=0\\ \Rightarrow \frac{3}{2}p{v}^{\frac{1}{2}}=-v^{\frac{3}{2}}\frac{dp}{dv}\\ \Rightarrow \frac{3}{2}p=-\frac{dp}{\frac{dv}{v}}\\ \therefore Bulkmodulus=\frac{3p}{2}\end{array}$

Hence, option $C$ is the correct answer.