### Thermodynamics Question 1

#### Question 1 - 2024 (01 Feb Shift 1)

The pressure and volume of an ideal gas are related as $PV^{3 / 2}=K$ (Constant). The work done by gas when the gas is taken from state $A\left(P _1, V _1, T _1\right)$ to state $B\left(P _2, V _2, T _2\right)$ is :

[We modified question statement, In official NTA paper question was incomplete]

(1) $2\left(P _1 V _1-P _2 V _2\right)$

(2) $2\left(P _2 V _2-P _1 V _1\right)$

(3) $2\left(\sqrt{P _1} V _1-\sqrt{P _2} V _2\right)$

(4) $2\left(P _2 \sqrt{V _2}-P _1 \sqrt{V _1}\right)$

## Show Answer

#### Answer: (1)

#### Solution:

For $PV^{x}=$ constant

If work done by gas is asked then

$W=\frac{nR \Delta T}{1-x}$

Here $x=\frac{3}{2}$

$\therefore W=\frac{P _2 V _2-P _1 V _1}{-\frac{1}{2}}$

$=2\left(P _1 V _1-P _2 V _2\right)$

Option (1) is correct

If work done by external is asked then $W=-2\left(P _1 V _1-P _2 V _2\right) \ldots \ldots$ Option (2) is correct