Question: 31: An ideal gas follows a process described by the equation PV^{2}=C from the initial \left(P_{1},V_{1},T_{1}\right) to final \left(P_{2},V_{2},T_{2}\right) thermodynamics states, where C is a constant. Then :
(1) If P_{1}>P_{2} then T_{1}<T_{2}
(2) If V_{2}>V_{1} then T_{2}>T_{1}
(3) If V_{2}>V_{1} then T_{2}<T_{1}
(4) If P_{1}>P_{2} then V_{1}>V_{2}
Answer: Option (3)
Explanation:
For an ideal gas, the equation of state is given by PV=nRT.
From the given process equation PV^{2}=C,
pressure can be written as P=\frac{C}{V^{2}}.
Substituting this value of pressure into the ideal gas equation,
we get \frac{C}{V^{2}}\cdot V=nRT.
Simplifying, \frac{C}{V}=nRT.
This shows that temperature is inversely proportional to volume,
i.e. T\propto\frac{1}{V}.
Therefore, if the volume increases from V_{1} to V_{2}
such that V_{2}>V_{1}, the temperature must decrease, giving
T_{2}<T_{1}Hence, the correct option is (3).