During an adiabatic process, if the pressure of a gas is found to be proportional to the cube of its absolute temperature, then the ratio of \fracC_PC_V for the gas is :

Heat and Thermodynamics PYQ — JEE Main 2024 · 5 Apr · Shift 2 · Q65 with Solution

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Correct answer: B

For an ideal gas undergoing an adiabatic process, $PV^{\gamma}=\text{constant}$ where $\gamma=\frac{C_P}{C_V}.$

Also, from the ideal gas equation, $PV=nRT \quad \Rightarrow \quad P=\frac{nRT}{V}.$

From $PV^{\gamma}=K,$ we get $P=\frac{K}{V^{\gamma}}.$

Using the ideal gas law, $T \propto PV.$ Substitute $P \propto V^{-\gamma}$ : $T \propto V^{-\gamma} \cdot V = V^{1-\gamma}.$

So, $V \propto T^{\frac{1}{1-\gamma}}.$

Now,

= T^{-\frac{\gamma}{1-\gamma}} = T^{\frac{\gamma}{\gamma-1}}.$$ Hence, for an adiabatic process, $$P \propto T^{\frac{\gamma}{\gamma-1}}.$$ 3. **Compare with the given condition** It is given that $$P \propto T^3.$$ Therefore, $$\frac{\gamma}{\gamma-1}=3.$$ 4. **Solve for $\gamma$** $$\gamma = 3(\gamma-1)$$ $$\gamma = 3\gamma - 3$$ $$2\gamma = 3$$ $$\gamma = \frac{3}{2}.$$ Thus, $$\frac{C_P}{C_V}=\frac{3}{2}.$$ 5. **Check options** - A: $\frac{5}{3}$ ❌ - B: $\frac{3}{2}$ ✅ - C: $\frac{7}{5}$ ❌ - D: $\frac{9}{7}$ ❌ So the correct option is **B**.