1. ⇒ (MHT CET 2023 12th May Evening Shift )

For polyatomic gases, the ratio of molar specific heat at constant pressure to constant volume is ( $f =$ degrees of freedom)

A. $\frac{2 + f}{3 + f}$

B. $\frac{3 + f}{2 + f}$

C. $\frac{3 + f}{4 + f}$

D. $\frac{4 + f}{3 + f}$

Correct Option is (D)

For polyatomic gases, molar-specific heat at constant volume is $C_{V} = (3 + f) R$ and Molar-specific heat at constant pressure is $C_{P} = (4 + f) R$

$∴ \frac{C_{p}}{C_{V}} = \frac{4 + f}{3 + f}$

2. ⇒ (MHT CET 2023 12th May Morning Shift )

Let $γ_{1}$ be the ratio of molar specific heat at constant pressure and molar specific heat at constant volume of a monoatomic gas and $γ_{2}$ be the similar ratio of diatomic gas. Considering the diatomic gas molecule as a rigid rotator, the ratio $\frac{γ_{2}}{γ_{1}}$ is

A. $\frac{37}{21}$

B. $\frac{27}{35}$

C. $\frac{21}{25}$

D. $\frac{35}{27}$

Correct Option is (C)

For monoatomic gas,

$γ_{1} = \frac{5}{3}$

For rigid diatomic gas,

$\begin{aligned} γ_{2} & = \frac{7}{5} \\ ∴ \frac{γ_{2}}{γ_{1}} & = \frac{7}{5} \times \frac{3}{5} = \frac{21}{25} \end{aligned}$

3. ⇒ (MHT CET 2023 12th May Morning Shift )

The molar specific heat of an ideal gas at constant pressure and constant volume is $C_{p}$ and $C_{v}$ respectively. If $R$ is universal gas constant and $γ = \frac{C_{p}}{C_{v}}$ then $C_{v} =$

A. $\frac{1 - γ}{1 + γ}$

B. $\frac{1 + γ}{1 - γ}$

C. $\frac{γ - 1}{R}$

D. $\frac{R}{γ - 1}$

Correct Option is (D)

$C_{p} - C_{v} = R$

Dividing both the sides by $C_{v}$ ,

$\begin{array}{lll} ∴ & γ - 1 = \frac{R}{C_{v}} & \dots . (∵ \frac{C_{p}}{C_{v}} = γ) \\ ∴ & C_{v} = \frac{R}{γ - 1} \end{array}$

4. ⇒ (MHT CET 2023 11th May Evening Shift )

For a gas, $\frac{R}{C_{v}} = 0 \cdot 4$ , where $R$ is universal gas constant and $C_{v}$ is molar specific heat at constant volume. The gas is made up of molecules which are

A. rigid diatomic

B. monoatomic

C. non-rigid diatomic

D. polyatomic

Correct Option is (A)

$\begin{aligned} Given: \frac{R}{C_{V}} = 0.4 \\ C_{V} = \frac{R}{0.4} = \frac{5 R}{2} \\ C_{P} = C_{V} + R \\ ∴ C_{P} = \frac{7 R}{2} \\ γ = \frac{C_{P}}{C_{v}} = \frac{\frac{7}{5}}{\frac{5}{2}} \\ γ = \frac{7}{5} \end{aligned}$

$∴$ The gas is made up of rigid diatomic molecules.

5. ⇒ (MHT CET 2023 9th May Morning Shift )

For a gas having ' $X$ ' degrees of freedom, ' $γ$ ' is ( $γ =$ ratio of specific heats $= C_{P} / C_{V}$ )

A. $\frac{1 + X}{2}$

B. $1 + \frac{X}{2}$

C. $1 + \frac{2}{x}$

D. $1 + \frac{1}{x}$

Correct Option is (C)

$γ$ and degrees of freedom is related by

$γ = \frac{f + 2}{f}$

Where $f$ is the number of degrees of freedoms.

Given $f = X$ ,

$∴ γ = \frac{X + 2}{X} = 1 + \frac{2}{X}$

⇒Kinetic Theory of Gases