NEET Behaviour of Perfect Gas and Kinetic Theory — Set 13

Question 1

The rms speed of a gas is 450 m/s at 225 K. At what temperature will the rms speed be 900 m/s?

Accepted Answer

$v_{\text{rms}} \propto \sqrt{T}$, $\frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}}$. $\frac{900}{450} = \sqrt{\frac{T_2}{225}}$, $2 = \sqrt{\frac{T_2}{225}}$. Square both sides: $4 = \frac{T_2}{225}$, $T_2 = 900 \, \text{K}$.

Question 2

A gas mixture has equal numbers of oxygen and argon molecules at 300 K. What is the ratio of their rms speeds? (Molecular mass: $O_2 = 32 \, \text{u}$, Ar = 39.9 u)

Accepted Answer

$v_{\text{rms}} \propto \frac{1}{\sqrt{m}}$, $\frac{v_{\text{O}_2}}{v_{\text{Ar}}} = \sqrt{\frac{m_{\text{Ar}}}{m_{\text{O}_2}}}$. $\frac{v_{\text{O}_2}}{v_{\text{Ar}}} = \sqrt{\frac{39.9}{32}} \approx \sqrt{1.247} \approx 1.117$.

Question 3

What is the total internal energy of 0.7 moles of a diatomic gas at 200 K with no vibrational modes? ($R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}$)

Accepted Answer

Diatomic gas: 5 degrees of freedom, $U = \frac{5}{2} \mu R T$. $U = \frac{5}{2} \times 0.7 \times 8.31 \times 200 = 2908.5 \, \text{J} \approx 2.91 \, \text{kJ}$.

NEET Physics: Behaviour of Perfect Gas and Kinetic Theory — Practice Set 13

Q1. The rms speed of a gas is 450 m/s at 225 K. At what temperature will the rms speed be 900 m/s?

Q2. A gas mixture has equal numbers of oxygen and argon molecules at 300 K. What is the ratio of their rms speeds? (Molecular mass: \(O_2 = 32 \, \text{u}\), Ar = 39.9 u)

Q3. What is the total internal energy of 0.7 moles of a diatomic gas at 200 K with no vibrational modes? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q4. A gas mixture contains 12 g of helium and 28 g of nitrogen. What is the ratio of their partial pressures?

Q5. The rms speed of argon molecules is 430 m/s at 300 K. What is the rms speed of nitrogen molecules at the same temperature? (Atomic mass: Ar = 39.9 u, \(N_2 = 28 \, \text{u}\))

Q6. How much heat is required to raise the temperature of 0.25 moles of a diatomic gas by 20 K at constant volume, with no vibrational modes? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q7. A gas has a \(C_p\) of \(29.8 \, \text{J mol}^{-1} \text{K}^{-1}\). What is the ratio of specific heats (\(\gamma\))? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q8. What is the average translational kinetic energy of a helium atom at 1000 K? (\(k_B = 1.38 \times 10^{-23} \, \text{J K}^{-1}\))

Q9. A solid has a molar specific heat capacity of \(25 \, \text{J mol}^{-1} \text{K}^{-1}\) at room temperature. How many degrees of freedom does each atom have?

Q10. What is the molar specific heat at constant volume (\(C_v\)) for a diatomic gas with no vibrational modes? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))