NEET Behaviour of Perfect Gas and Kinetic Theory — Set 2

Question 1

A gas has a $C_v$ of $12.7 \, \text{J mol}^{-1} \text{K}^{-1}$. What is the ratio of specific heats ($\gamma$)? ($R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}$)

Accepted Answer

$C_p = C_v + R = 12.7 + 8.31 = 21.01 \, \text{J mol}^{-1} \text{K}^{-1}$. $\gamma = \frac{C_p}{C_v} = \frac{21.01}{12.7} \approx 1.65$.

Question 2

A gas mixture has equal numbers of nitrogen and oxygen molecules at 300 K. What is the ratio of their rms speeds? (Molecular mass: $N_2 = 28 \, \text{u}$, $O_2 = 32 \, \text{u}$)

Accepted Answer

$v_{\text{rms}} \propto \frac{1}{\sqrt{m}}$, $\frac{v_{\text{N}_2}}{v_{\text{O}_2}} = \sqrt{\frac{m_{\text{O}_2}}{m_{\text{N}_2}}}$. $\frac{v_{\text{N}_2}}{v_{\text{O}_2}} = \sqrt{\frac{32}{28}} \approx \sqrt{1.1429} \approx 1.069$.

Question 3

The mean free path of a gas molecule is $8 \times 10^{-7} \, \text{m}$ at 1 atm. What will it be at 0.25 atm if temperature remains constant?

Accepted Answer

$l \propto \frac{1}{n}$, $n \propto P$. If $P$ reduces to $\frac{1}{4}$, $n$ reduces to $\frac{1}{4}$, $l$ increases 4 times. New $l = 8 \times 10^{-7} \times 4 = 3.2 \times 10^{-6} \, \text{m}$.

NEET Physics: Behaviour of Perfect Gas and Kinetic Theory — Practice Set 2

Q1. A gas has a \(C_v\) of \(12.7 \, \text{J mol}^{-1} \text{K}^{-1}\). What is the ratio of specific heats (\(\gamma\))? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q2. A gas mixture has equal numbers of nitrogen and oxygen molecules at 300 K. What is the ratio of their rms speeds? (Molecular mass: \(N_2 = 28 \, \text{u}\), \(O_2 = 32 \, \text{u}\))

Q3. The mean free path of a gas molecule is \(8 \times 10^{-7} \, \text{m}\) at 1 atm. What will it be at 0.25 atm if temperature remains constant?

Q4. A solid has a molar specific heat capacity of \(26.5 \, \text{J mol}^{-1} \text{K}^{-1}\). How many degrees of freedom per atom does it have? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q5. A gas has a \(C_p\) of \(35.4 \, \text{J mol}^{-1} \text{K}^{-1}\). What is the ratio of specific heats (\(\gamma\))? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q6. The rms speed of argon molecules is 430 m/s at 300 K. What is the rms speed of neon molecules at the same temperature? (Atomic mass: Ar = 39.9 u, Ne = 20.2 u)

Q7. A solid has a molar specific heat capacity of \(24.4 \, \text{J mol}^{-1} \text{K}^{-1}\). How many degrees of freedom per atom does it have? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q8. A gas at 1 atm and 273 K has a volume of 11.2 litres. If the temperature increases to 546 K at constant pressure, what is the new volume?

Q9. The rms speed of a gas is 500 m/s at 250 K. At what temperature will the rms speed be 1000 m/s?

Q10. The mean free path of a gas molecule is \(2.0 \times 10^{-6} \, \text{m}\) at 0.1 atm. What will it be at 0.4 atm if temperature remains constant?