Question

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

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

Answer 1

Answer: (C)

$\textit{PT}^{2}=\text{constant}$
$\therefore \left(\frac{n \textit{RT}}{\textit{V}}\right)\left(\textit{T}\right)^{2}=\text{constant}$

or, $\textit{T}^{3}\textit{V}^{- 1}=\text{constant}$
Differentiating the equation, we get
$\frac{3 \textit{T}^{2}}{\textit{V}}\cdot \text{d}\textit{T}-\frac{\textit{T}^{3}}{\textit{V}^{2}}\text{d}\textit{V}=0$
or, $3\text{d}\textit{T}=\frac{\textit{T}}{\textit{V}}\cdot \text{d}\textit{V}$ ..........(i)
From the equation, $\text{d}\textit{V}=\textit{V}\gamma \text{d}\textit{T}$
$\gamma =\text{Coefficient of volume expansion of gas}=\frac{\text{d} \textit{V}}{\textit{V} \cdot \text{d} \textit{T}}$
From Eq.(i) $\gamma =\frac{\text{d} \textit{V}}{\textit{V} \cdot \text{d} \textit{T}}=\frac{3}{\textit{T}}$
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