MHT CET · Physics · Kinetic Theory of Gases

Four moles of hydrogen, two moles of helium and one mole of water vapour form an ideal gas mixture. \(\left[C_v\right.\) for hydrogen \(=\frac{5}{2} R, \quad C_v\) for helium \(=\frac{3}{2} R, \quad C_v \quad\) for water vapour \(\left.=3 R\right]\) What is the molar specific heat at constant pressure of the mixture?

A \(\frac{11}{3} \mathrm{R}\)
B \(\frac{23}{7} \mathrm{R}\)
C \(\frac{16}{7} \mathrm{R}\)
D \(\frac{23}{3} \mathrm{R}\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\frac{23}{7} \mathrm{R}\)

Step-by-step Solution

Detailed explanation

\(C_p=C_v+R\)
\(\therefore \quad \mathrm{C}_{\mathrm{p}}\) for hydrogen, \(\mathrm{C}_{\mathrm{p}_1}=\frac{7}{2} \mathrm{R}\)
\(\mathrm{C}_{\mathrm{p}}\) for helium, \(\mathrm{C}_{\mathrm{p}_2}=\frac{5}{2} \mathrm{R}\)
\(\mathrm{C}_{\mathrm{p}}\) for water vapour, \(\mathrm{C}_{\mathrm{p}_3}=4 \mathrm{R}\)
Also, \(\mathrm{n}_1=4, \mathrm{n}_2=2\) and \(\mathrm{n}_3=1\)
\(\begin{aligned}
C_p \text { of mixture } & =\frac{n_1 C_{p_1}+n_2 C_{p_2}+n_3 C_{p_3}}{n_1+n_2+n_3} \\
& =\frac{4 \times \frac{7 R}{2}+2 \times \frac{5 R}{2}+4 R}{4+2+1} \\
& =\frac{14 R+5 R+4 R}{7}=\frac{23 R}{7}
\end{aligned}\)

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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