Question

An ideal gas undergoes a thermodynamic process in which the work done by the gas is the same as the increase in the internal energy of the gas. The molar heat capacity of the gas for the process is
(Given, $\gamma =1.5$ for the gas)

• A. $4R$
• B. $2R$
• C. $3R$
• D. $5R$

Answer 1

The correct option is A $4R$

Given:
$\gamma =1.5$
From the first law of thermodynamics,
$\Delta Q=\Delta U+W$
For this process,
$W=\Delta U$ (Given)
$\Rightarrow \Delta Q=\Delta U+\Delta U$
$\Rightarrow \Delta Q=2\Delta U$ ...........(1)
Let $C$ be the molar heat capacity of gas for the given process.
We know that, $\Delta Q=nC\Delta T$
and $\Delta U=n{C}_V\Delta T$
Substituting in the equation (1),
$\Rightarrow nC\Delta T=2n{C}_V\Delta T$
$\Rightarrow C=2C_V$
Using, $C_V=\frac{R}{\gamma -1}=\frac{R}{1.5-1}=2R$
$\Rightarrow C=2\left(2R\right)$
$\therefore C=4R$