Chemistry - Ideal Gas Question with Solution | TestHub

Question Explanation : Match gases with their molar heat capacity $\left(C_{v,m}\right)$ values assuming active vibrational modes.

Concept: Degrees of freedom theory: $C_{n,m}=\left(\frac{f_{\text{nees }}+f_{\text{net }}}{2}+f_{\text{vib }}\right)R$.

Solution:

When vibrational modes are active, each vibrational degree of freedom contributes R to the molar heat capacity.

(P) He: A monoatomic gas has 3 translational degrees of freedom only. $C_{v,m}=\frac{3}{2}R=1.5R$. (P → 4)

(Q) O₂: A diatomic molecule has 3 translational, 2 rotational, and 1 vibrational mode ($3N-5=1$). $C_{v,m}=\left(\frac{3+2}{2}+1\right)R=(2.5+1)R=3.5R$. (Q → 3)

(R) SO₂: A non-linear triatomic molecule ($N=3$) has 3 translational, 3 rotational, and 3 vibrational modes ($3N-6=3$). $C_{v,m}=\left(\frac{3+3}{2}+3\right)R=(3+3)R=6R$. (R → 1)

(S) CH₄: A non-linear polyatomic molecule ($N=5$) has 3 translational, 3 rotational, and 9 vibrational modes ($3N-6=9$). $C_{v,m}=\left(\frac{3+3}{2}+9\right)R=(3+9)R=12R$. (S → 2)

Matching these results, we get the sequence: P → 4, Q → 3, R → 1, S → 2.

Final Answer: Option (A)