6. (JEE Main 2022 (Online) 27th July Evening Shift )

Which statements are correct about degrees of freedom ?

(A) A molecule with n degrees of freedom has n $^{2}$ different ways of storing energy.

(B) Each degree of freedom is associated with $\frac{1}{2}$ RT average energy per mole.

(C) A monatomic gas molecule has 1 rotational degree of freedom where as diatomic molecule has 2 rotational degrees of freedom.

(D) ${CH}_{4}$ has a total of 6 degrees of freedom.

Choose the correct answer from the options given below :

(A) (B) and (C) only

(B) (B) and (D) only

(C) (A) and (B) only

(D) (C) and (D) only

Correct answer is (B)

Statement A is incorrect, statement B is correct by equipartition of energy. Statement C is incorrect as monoatomic does not have any rotational degree of freedom and CH₄ is a polyatomic gas so it has 6 degree of freedom. So only B and D are correct.

7. (JEE Main 2022 (Online) 26th July Evening Shift )

A gas has $n$ degrees of freedom. The ratio of specific heat of gas at constant volume to the specific heat of gas at constant pressure will be :

(A) $\frac{n}{n + 2}$

(B) $\frac{n + 2}{n}$

(C) $\frac{n}{2 n + 2}$

(D) $\frac{n}{n - 2}$

Correct answer is (A)

$C_{V} = \frac{n R}{2}$

And $C_{P} = \frac{n R}{2} + R$

$\Rightarrow \frac{C_{V}}{C_{P}} = \frac{\frac{n R}{2}}{\frac{n R}{2} + R} = \frac{n}{n + 2}$

8. (JEE Main 2022 (Online) 26th June Morning Shift )

A thermally insulated vessel contains an ideal gas of molecular mass M and ratio of specific heats 1.4. Vessel is moving with speed v and is suddenly brought to rest. Assuming no heat is lost to the surrounding and vessel temperature of the gas increases by :

(R = universal gas constant)

(A) $\frac{M v^{2}}{7 R}$

(B) $\frac{M v^{2}}{5 R}$

(C) 2 $\frac{M v^{2}}{7 R}$

(D) 7 $\frac{M v^{2}}{5 R}$

Correct answer is (B)

Let there be n moles of gas

E_loss = E_gain

$\frac{1}{2} (n M) v^{2} = n C_{v} Δ T$

$\frac{1}{2} M v^{2} = C_{v} Δ T$

here, $γ = 1.4 = \frac{7}{5}$ i.e. diatomic gas

$∴$ $C_{v} = \frac{5 R}{2}$

Now, $\frac{1}{2} M v^{2} = \frac{5 R}{2} Δ T$

$Δ T = \frac{M v^{2}}{5 R}$

9. (JEE Main 2022 (Online) 25th June Evening Shift )

The ratio of specific heats $(\frac{C_{P}}{C_{V}})$ in terms of degree of freedom (f) is given by :

(A) $(1 + \frac{f}{3})$

(B) $(1 + \frac{2}{f})$

(C) $(1 + \frac{f}{2})$

(D) $(1 + \frac{1}{f})$

Correct answer is (B)

$\frac{C_{P}}{C_{V}} = γ$

$C_{V} = (\frac{f}{2}) R$ and $C_{P} - C_{V} = R$

$\Rightarrow \frac{C_{P}}{C} = \frac{1 + f / 2}{f / 2} = 1 + \frac{2}{f}$

10. (JEE Main 2022 (Online) 29th July Morning Shift )

One mole of a monoatomic gas is mixed with three moles of a diatomic gas. The molecular specific heat of mixture at constant volume is $\frac{α^{2}}{4} R J / mol K$ ; then the value of $α$ will be _________. (Assume that the given diatomic gas has no vibrational mode).

Correct answer is (3)

$C_{V} = \frac{f}{2} R$

total degree of freedoms

$= 1 \times 3 + 3 \times 5 = 18$

$\frac{α^{2}}{4} = \frac{18}{2 n} = \frac{18}{2 \times 4}$

$\Rightarrow α^{2} = 9$

$α = 3$

Kinetic Theory of Gases