Question

An ideal gas mixture filled inside a balloon expands according to the relation ${\mathrm{PV}}^{\frac{2}{3}}=\mathrm{constant}$. The temperature inside the balloon is:

• A. increasing
• B. decreasing
• C. constant
• D. cannot be defined

Answer 1

The correct option is A

increasing

Explanation for the correct option

Step 1: Given

An ideal gas that is expanding.

Relation between pressure and volume: ${\mathrm{PV}}^{\frac{2}{3}}=\mathrm{constant}$

Step 2: Formula used

The ideal gas equation: $\mathrm{PV}=\mathrm{nRT}$, where

$\mathrm{P}=\mathrm{Pressure}$

$\mathrm{V}=\mathrm{Volume}$

$\mathrm{n}=\mathrm{Number}\mathrm{of}\mathrm{moles}$

$\mathrm{R}=\mathrm{Gas}\mathrm{constant}$

$\mathrm{T}=\mathrm{Temperature}$

Step 3: Calculations

Using the ideal gas equation $\mathrm{P}=\frac{\mathrm{nRT}}{\mathrm{V}}$

Putting the value of $\mathrm{P}$ in the relation ${\mathrm{PV}}^{\frac{2}{3}}=\mathrm{constant}$
$\Rightarrow \left(\frac{\mathrm{nRT}}{\mathrm{V}}\right){\mathrm{V}}^{\frac{2}{3}}=\mathrm{k}$ where $\mathrm{k}$ is constant.

$\Rightarrow \frac{\mathrm{T}}{{\mathrm{V}}^{{\displaystyle \frac{1}{3}}}}=\mathrm{C}$ where $\mathrm{C}=\frac{\mathrm{k}}{\mathrm{nR}}$is a constant.

$\Rightarrow \mathrm{T}={\mathrm{V}}^{\frac{1}{3}}\mathrm{C}$

Therefore the temperature of the gas is increasing.

Hence this option is correct.