Question

An ideal gas is expanding such that $P{T}^2$ = 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)

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

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