Question

A gas expands adiabatically at constant pressure, such that its temperature $T\propto \frac{1}{\sqrt{V}}$. The value of $C_p/C_V$ of the gas is:

• A. 1.3
• B. 1.5
• C. 1.67
• D. 2

Answer 1

The correct option is B 1.5

Key Idea Sudden expansion is adiabatic operation. An adiabatic relation between p and V is
$p{V}^{\gamma }=k$, a constant.
A change in pressure and volume of a gas in which temperature also changes is called an adiabatic change. In such a change, no heat is allowed to enter into or escape from the gas (ie, there is no exchange of heat between the gas and its surroundings).
For adiabatic expansion, we have the formula
$p{V}^{\gamma }=$ constant ...(i)
Gas equation is,
$pV=RT$
$\Rightarrow p=\frac{RT}{V}$ ...(ii)
From Eqs. (i) and (ii), we obtain
$\left(\frac{RT}{V}\right)V^{\gamma }=$ constant
$\Rightarrow T{V}^{\gamma -1}=$constant ...(iii)
But $T\propto \frac{1}{\sqrt{V}}$ (given)
as $T{V}^{1/2}=$ constant ...(iv)
Thus, using Eqs. (iii) and (iv) together, we get
$\gamma -1=\frac{1}{2}$
or $\gamma =\frac{3}{2}=1.5$
$\Rightarrow \frac{C_p}{C_v}=1.5$