Thermodynamics Question 1
Question 1 - 2024 (01 Feb Shift 1)
The pressure and volume of an ideal gas are related as $\mathrm{PV}^{3 / 2}=\mathrm{K}$ (Constant). The work done by gas when the gas is taken from state $A\left(P_{1}, V_{1}, T_{1}\right)$ to state $\mathrm{B}\left(\mathrm{P}{2}, \mathrm{~V}{2}, \mathrm{~T}_{2}\right)$ is :
[We modified question statement, In official NTA paper question was incomplete]
(1) $2\left(\mathrm{P}{1} \mathrm{~V}{1}-\mathrm{P}{2} \mathrm{~V}{2}\right)$
(2) $2\left(\mathrm{P}{2} \mathrm{~V}{2}-\mathrm{P}{1} \mathrm{~V}{1}\right)$
(3) $2\left(\sqrt{\mathrm{P}{1}} \mathrm{~V}{1}-\sqrt{\mathrm{P}{2}} \mathrm{~V}{2}\right)$
(4) $2\left(P_{2} \sqrt{V_{2}}-P_{1} \sqrt{V_{1}}\right)$
Show Answer
Answer: (1)
Solution:
For $\mathrm{PV}^{\mathrm{x}}=$ constant
If work done by gas is asked then
$\mathrm{W}=\frac{\mathrm{nR} \Delta \mathrm{T}}{1-\mathrm{x}}$
Here $\mathrm{x}=\frac{3}{2}$
$\therefore \mathrm{W}=\frac{\mathrm{P}{2} \mathrm{~V}{2}-\mathrm{P}{1} \mathrm{~V}{1}}{-\frac{1}{2}}$
$=2\left(\mathrm{P}{1} \mathrm{~V}{1}-\mathrm{P}{2} \mathrm{~V}{2}\right)$
Option (1) is correct
If work done by external is asked then $\mathrm{W}=-2\left(\mathrm{P}{1} \mathrm{~V}{1}-\mathrm{P}{2} \mathrm{~V}{2}\right) \ldots \ldots$ Option (2) is correct
