Question

The ratio of the isothermal elasticity of gas to its adiabatic elasticity is :

• A. $\gamma$
• B. $\frac{1}{\gamma }$
• C. $1-\gamma$
• D. $\frac{1}{1-\gamma }$

Answer 1

The correct option is B $\frac{1}{\gamma }$

Elasticity is also defined using bulk modulus.
Bulk modulus is given by $K=-V\frac{dP}{dV}$
For an isothermal process, temperature is constant
$PV=$constant
$d\left(PV\right)=0$
$PdV+dPV=0$
$P=-V\frac{dP}{dV}$
So, for an isothermal process, buld modulus$K_T=P$
For an adiabatic process,
$P{V}^{\gamma }=$constant
$d\left(P{V}^{\gamma }\right)=0$
$dP.V^{\gamma }+P\left(\gamma V^{\gamma -1}\right)dV=0$
$dP.V^{\gamma +1}+\gamma P{V}^{\gamma }dV=0$
$\gamma P=-V\frac{dP}{dV}$
So, for an adiabatic process, bulk modulus$K_s=\gamma P$
$\therefore \frac{K_T}{K_s}=\frac{P}{\gamma P}=\frac{1}{\gamma }$
Option B.