NEET Behaviour of Perfect Gas and Kinetic Theory — Set 14

Question 1

The rms speed of oxygen molecules is 482 m/s at 300 K. What is the rms speed of methane molecules at the same temperature? (Molecular mass: $O_2 = 32 \, \text{u}$, $CH_4 = 16 \, \text{u}$)

Accepted Answer

$v_{\text{rms}} \propto \frac{1}{\sqrt{m}}$, $\frac{v_{\text{CH}_4}}{v_{\text{O}_2}} = \sqrt{\frac{m_{\text{O}_2}}{m_{\text{CH}_4}}}$. $\frac{v_{\text{CH}_4}}{482} = \sqrt{\frac{32}{16}} = \sqrt{2} \approx 1.414$. $v_{\text{CH}_4} = 482 \times 1.414 \approx 682 \, \text{m/s}$.

Question 2

A gas mixture has equal masses of helium and nitrogen at 300 K. What is the ratio of their rms speeds? (Atomic mass: He = 4 u, $N_2 = 28 \, \text{u}$)

Accepted Answer

$v_{\text{rms}} \propto \frac{1}{\sqrt{m}}$, $\frac{v_{\text{He}}}{v_{\text{N}_2}} = \sqrt{\frac{m_{\text{N}_2}}{m_{\text{He}}}}$. $\frac{v_{\text{He}}}{v_{\text{N}_2}} = \sqrt{\frac{28}{4}} = \sqrt{7} \approx 2.645$.

Question 3

What is the volume of 0.5 moles of an ideal gas at 2 atm and 627°C? ($R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}$)

Accepted Answer

$PV = \mu R T$, $V = \frac{\mu R T}{P}$. $T = 627 + 273 = 900 \, \text{K}$, $P = 2 \times 1.01 \times 10^5 = 2.02 \times 10^5 \, \text{Pa}$. $V = \frac{0.5 \times 8.31 \times 900}{2.02 \times 10^5} = 1.854 \times 10^{-2} \, \text{m}^3 \approx 18.54 \, \text{litres}$.

NEET Physics: Behaviour of Perfect Gas and Kinetic Theory — Practice Set 14

Q1. The rms speed of oxygen molecules is 482 m/s at 300 K. What is the rms speed of methane molecules at the same temperature? (Molecular mass: \(O_2 = 32 \, \text{u}\), \(CH_4 = 16 \, \text{u}\))

Q2. A gas mixture has equal masses of helium and nitrogen at 300 K. What is the ratio of their rms speeds? (Atomic mass: He = 4 u, \(N_2 = 28 \, \text{u}\))

Q3. What is the volume of 0.5 moles of an ideal gas at 2 atm and 627°C? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q4. The rms speed of nitrogen molecules is 516 m/s at 300 K. What will it be at 600 K? (Molecular mass of \(N_2 = 28 \, \text{u}\))

Q5. A gas at 1.5 atm and 300 K has a volume of 18 litres. If the pressure decreases to 0.75 atm at constant temperature, what is the new volume?

Q6. A gas occupies 89.6 litres at STP. How many moles are present? (Molar volume at STP = 22.4 litres)

Q7. What is the temperature at which the average translational kinetic energy of a gas molecule is \(1.035 \times 10^{-20} \, \text{J}\)? (\(k_B = 1.38 \times 10^{-23} \, \text{J K}^{-1}\))

Q8. The mean free path of a gas molecule is \(3.0 \times 10^{-6} \, \text{m}\) at 0.25 atm. What will it be at 1 atm if temperature remains constant?

Q9. What is the total internal energy of 2 moles of a monatomic gas at 400 K? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q10. A gas has a \(C_p\) of \(35.4 \, \text{J mol}^{-1} \text{K}^{-1}\). What is its \(C_v\)? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))