MHT CET · Physics · Kinetic Theory of Gases

For an ideal gas the density of the gas is \(\rho_0\) when temperature and pressure of the gas are \(T_0\) and \(\mathrm{P}_0\) respectively. When the temperature of the gas is \(2 \mathrm{~T}_0\), its pressure will be \(3 \mathrm{P}_0\). The new density will be

A \(\frac{3}{2} \rho_0\)
B \(\frac{4}{3} \rho_0\)
C \(\frac{3}{4} \rho_0\)
D \(\frac{2}{3} \rho_0\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(\frac{3}{2} \rho_0\)

Step-by-step Solution

Detailed explanation

Density \(\propto \mathrm{P} / \mathrm{T}\)
So, \(\frac{\mathrm{d}_2}{\mathrm{~d}_1}=\frac{\mathrm{P}_2}{\mathrm{~T}_2} \times \frac{\mathrm{T}_1}{\mathrm{P}_{\mathrm{t}}}\)
\(\therefore \quad\) The new density is:
\(
\begin{aligned}
& \mathrm{d}_2=\rho_0 \times \frac{3 \mathrm{P}_0}{2 \mathrm{~T}_0} \times \frac{\mathrm{T}_0}{\mathrm{P}_0} \\
& \mathrm{~d}_2=\frac{3}{2} \rho_0
\end{aligned}
\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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