Question

The temperature of $n$ moles of an ideal gas is increased from $T$ to $4T$ through a process for which pressure $P = a T^{- 1}$ , where $a$ is a constant. The work done by the gas is

• A. $nRT$
• B. $4nRT$
• C. $2nRT$
• D. $6nRT$

Answer 1

Answer: (D)

According to ideal gas equation
$P V = n R T$
$\text{or} V = \frac{ n R T ⁡}{ P ⁡}$
$\because P = \frac{a}{ T ⁡} \left(\text{Given}\right) \ldots \left(\text{i}\right)$
$\because V = \frac{n R ⁡ T ⁡^{2}}{a}$
$⇒ \, \textit{dV} = \frac{2nRT}{\textit{a}} \textit{dT}$
Work done by the gas, $\text{dW = PdV}$
$\text{or } \textit{W} = \int\limits _{ T }^{4 T ⁡} \frac{a}{ T ⁡} \frac{2 n R ⁡ T ⁡}{a} d T ⁡ \ldots \left(\text{Using (i) and (ii)}\right)$
$= \left[2 n R ⁡ T ⁡\right]_{ T ⁡}^{4 T ⁡} = 6 n R T ⁡$