Question

One mole of an ideal diatomic gas undergoes a transition from $\text{A to B}$ along a path $\text{AB}$ as shown in the figure. The change in internal energy of the gas during the transition is:

• A. $12\text{kJ}$
• B. $-12\text{kJ}$
• C. $20\text{kJ}$
• D. $-20\text{kJ}$

Answer 1

The correct option is D $-20\text{kJ}$

The change in internal energy for diatomic gas is given by,

$\Delta U=n{C}_V\Delta T=n{C}_V(T_B-T_A)$

$n\to$ no of mole, Here $n=1$
$C_V\to$ Specific heat capacity at constant volume
$\Delta T\to$ change in temperature

For a diatomic gas, $C_V=\frac{5}{2}R$

From ideal gas equation, $PV=nRT$

$T_A=\frac{P_A{V}_A}{R};T_B=\frac{P_B{V}_B}{R}$

$\Rightarrow \Delta U=1\times \frac{5}{2}\times R\times (\frac{P_B{V}_B}{R}-\frac{P_A{V}_A}{R})$

$\Delta U=\frac{5}{2}(P_B{V}_B-P_A{V}_A)$

From the given graph,
$P_A=5\times {10}^3\text{Pa};V_A=4{\text{m}}^3$
$P_B=2\times {10}^3\text{Pa};V_B=6{\text{m}}^3$

$\Rightarrow \Delta U=\frac{5}{2}\left[\right(2\times {10}^3\times 6)-(5\times {10}^3\times 4\left)\right]$

$\Rightarrow \Delta U=-20\text{kJ}$

Hence, option $\left(\text{D}\right)$ is correct.