Question

An ideal gas is expanding such that ${\mathrm{PT}}^2=\mathrm{constant}.$ The coefficient of volume expansion of the gas is

• A. $\frac{1}{\mathrm{T}}$
• B. $\frac{2}{\mathrm{T}}$
• C. $\frac{3}{\mathrm{T}}$
• D. $\frac{4}{\mathrm{T}}$

Answer 1

The correct option is C

$\frac{3}{\mathrm{T}}$

Explanation for the correct option :

Option C: $\frac{3}{\mathrm{T}}$

Step 1: Given

An ideal gas that is expanding.

Relation between pressure and temperature: ${\mathrm{PT}}^2=\mathrm{constant}.$

Step 2: Formula used

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

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

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

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

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

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

Coefficient of the volume of expansion is calculated using the formula: $\frac{d\mathrm{V}}{\mathrm{V}d\mathrm{T}}=coefficientofvolumeexpansion$where

$\mathrm{V}=\mathrm{volume}$

$\mathrm{dV}=\mathrm{change}\mathrm{in}\mathrm{volume}$

$\mathrm{dT}=\mathrm{change}\mathrm{in}\mathrm{temperature}$

Step 3: Calculations

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

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

$\Rightarrow {\mathrm{nRT}}^3=\mathrm{V}\mathrm{k}...\left(\mathrm{i}\right)$

Differentiating both sides

$\Rightarrow 3{\mathrm{nRT}}^{2}d\mathrm{T}=d\mathrm{V}\mathrm{k}....\left(\mathrm{ii}\right)$

Dividing equation (ii) by equation (i)

$$\begin{gathered} \Rightarrow \frac{3{\mathrm{nRT}}^{2}d\mathrm{T}}{{\mathrm{nRT}}^3}=\frac{d\mathrm{Vk}}{\mathrm{Vk}} \\ \Rightarrow \frac{3d\mathrm{T}}{\mathrm{T}}=\frac{d\mathrm{V}}{\mathrm{V}} \\ \Rightarrow \frac{3}{\mathrm{T}}=\frac{d\mathrm{V}}{\mathrm{V}d\mathrm{T}} \end{gathered}$$

Using the formula for the coefficient of volume expansion,

$\frac{d\mathrm{V}}{\mathrm{V}d\mathrm{T}}=\frac{\mathit{3}}{\mathit{T}}$

Hence this option is correct.