Match the $\mathrm{C}_{p} / \mathrm{C}_{\mathrm{V}}$ ratio for ideal gases with different type of molecules: | Molecule Type | | $\mathbf{C}_{\mathbf{P}} \boldsymbol{/} \mathbf{C}_{\mathbf{V}}$ | |---|---|---| | (A) Monatomic | (I) | $\frac{7}{5}$ | | (B) Diatomic rigid molecules | (II) | $\frac{9}{7}$ | | (C) Diatomic non-rigid molecules | (III) | $\frac{4}{3}$ | | (D) Triatomic rigid molecules | (IV) | $\frac{5}{3}$ | Match the correct option?

(B) A $\rightarrow$ IV; $\mathrm{B} \rightarrow \mathrm{I}$; C → II; D → III

(A) A $\rightarrow$ III; B → IV; $\mathrm{C} \rightarrow \mathrm{II}$; D $\rightarrow 1$

(C) $\mathrm{A} \rightarrow \mathrm{II}$; B → III; $\mathrm{C} \rightarrow \mathrm{I}$; D → IV

(D) A → IV; B → II; $\mathrm{C} \rightarrow \mathrm{I}$; D → III

JEE Main · 2020 · 04 Sep 2020 · Shift 1 · Heat & Thermodynamics

Match the

C_{p} / C_{V}

ratio for ideal gases with different type of molecules:

Molecule Type		$C_{P} / C_{V}$
(A) Monatomic	(I)	$\frac{7}{5}$
(B) Diatomic rigid molecules	(II)	$\frac{9}{7}$
(C) Diatomic non-rigid molecules	(III)	$\frac{4}{3}$
(D) Triatomic rigid molecules	(IV)	$\frac{5}{3}$

Match the correct option?

Answer: (B)

The ratio

γ = C_{p} / C_{v}

is fixed by the number of active degrees of freedom through

C_{v} = \frac{f}{2} R

and

C_{p} = C_{v} + R

. A monatomic gas has only

3

translational modes, so

C_{v} = \frac{3}{2} R

C_{p} = \frac{5}{2} R

and

γ = \frac{5}{3}

. A rigid diatomic molecule adds

2

rotational modes, giving

C_{v} = \frac{5}{2} R

C_{p} = \frac{7}{2} R

and

γ = \frac{7}{5}

. A non-rigid diatomic molecule also has

2

vibrational modes, so

C_{v} = \frac{7}{2} R

C_{p} = \frac{9}{2} R

and

γ = \frac{9}{7}

. A rigid (non-linear) triatomic molecule has

3

rotational modes, so

C_{v} = 3 R

C_{p} = 4 R

and

γ = \frac{4}{3}

. Thus the monatomic gas pairs with

\frac{5}{3}

, the rigid diatomic with

\frac{7}{5}

, the non-rigid diatomic with

\frac{9}{7}

and the rigid triatomic with

\frac{4}{3}

JEE Main · 2020 · 04 Sep 2020 · Shift 1 · Single correct option