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1. ⇒ (AIPMT 2015 Cancelled Paper )

The ratio of the specific heats $\frac{C_{p}}{C_{v}} = γ$ in terms of degrees of freedom (n) is given by

(A) $(1 + \frac{2}{n})$

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

(C) $(1 + \frac{1}{n})$

(D) $(1 + \frac{n}{3})$

Correct answer is (A)

For n degrees of freedom, $C_{v} = \frac{n}{2} R$

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

$C_{p} = C_{v} + R = \frac{n}{2} R + R = (\frac{n}{2} + 1) R$

$γ = \frac{C_{p}}{C_{v}} = \frac{(\frac{n}{2} + 1) R}{\frac{n}{2} R} = \frac{n + 2}{n}$

$∴$ $γ = 1 + \frac{2}{n}$

2. ⇒ (AIPMT 2010 Mains )

If c_p and c_v denote the specific heats (per unit mass of an ideal gas of molecular weight M, then

(A) c_p $-$ c_v = R/M²

(B) c_p $-$ c_v = R

(C) c_p $-$ c_v = R/M

(D) c_p $-$ c_v = MR

Correct answer is (C)

C_v = molar specific heat of the ideal gas at constant volume

C_p = molar specific heat of the ideal gas at constant pressure,

C_p' = MC_p and C_v’ = MC_v

Also ${C_{p}}^{'} - {C_{v}}^{'} = R$

MC_p – MC_v = R

C_p – C_v = R/M

3. ⇒ (AIPMT 2006 )

The molar specific heat at constant pressure of an ideal gas is (7/2) R. The ratio of specific heat at constant pressure to that at constant volume is

(A) 9/7

(B) 7/5

(C) 8/7

(D) 5/7

Correct answer is (B)

$C_{P} = \frac{7}{2} R; C_{V} = C_{P} - R$

$= \frac{7}{2} R - R = \frac{5}{2} R$

$\frac{C_{P}}{C_{V}} = \frac{7 / 2 R}{5 / 2 R} = \frac{7}{5}$

4. ⇒ (AIPMT 2000 )

To find out degree of freedom, the expansion is

(A) $f = \frac{2}{γ - 1}$

(B) $f = \frac{γ + 1}{2}$

(C) $f = \frac{2}{γ + 1}$

(D) $f = \frac{1}{γ + 1}$

Correct answer is (A)

$γ = 1 + \frac{2}{f}$

where f is the degree of freedom

$∴$ $\frac{2}{f} = γ - 1 \Rightarrow f = \frac{2}{γ - 1}$

⇒Kinetic Theory of Gases

Topic 04: Degree of Freedom and Specific Heat