NEET Behaviour of Perfect Gas and Kinetic Theory — Set 8

Question 1

The mean free path of air molecules at STP is $2.9 \times 10^{-7} \, \text{m}$. If temperature doubles at constant pressure, what is the new mean free path?

Accepted Answer

$l = \frac{1}{\sqrt{2} n \pi d^2}$, $n = \frac{N}{V}$, and $PV = Nk_B T$. At constant $P$, $V \propto T$, so $n \propto \frac{1}{T}$. If $T$ doubles, $n$ halves, $l$ doubles. New $l = 2 \times 2.9 \times 10^{-7} = 5.8 \times 10^{-7} \, \text{m} $.

Question 2

At what temperature is the rms speed of oxygen molecules 964 m/s? (Molecular mass of $O_2 = 32 \, \text{u}$, $k_B = 1.38 \times 10^{-23} \, \text{J K}^{-1}$)

Accepted Answer

$v_{\text{rms}} = \sqrt{\frac{3k_B T}{m}}$, $m = \frac{32 \times 10^{-3}}{6.02 \times 10^{23}} = 5.32 \times 10^{-26} \, \text{kg}$. $964^2 = \frac{3 \times 1.38 \times 10^{-23} \times T}{5.32 \times 10^{-26}}$, $T = \frac{9.29 \times 10^5 \times 5.32 \times 10^{-26}}{4.14 \times 10^{-23}} \approx 1194 \, \text{K}$.

Question 3

A mixture of 0.1 moles of helium and 0.9 moles of oxygen is at 500 K in a 30-litre container. What is the total pressure? ($R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}$)

Accepted Answer

$PV = \mu R T$, $P = \frac{\mu R T}{V}$. Total moles = $0.1 + 0.9 = 1.0$, $V = 30 \times 10^{-3} \, \text{m}^3$. $P = \frac{1.0 \times 8.31 \times 500}{30 \times 10^{-3}} = 1.385 \times 10^5 \, \text{Pa} \approx 1.39 \, \text{atm}$.

NEET Physics: Behaviour of Perfect Gas and Kinetic Theory — Practice Set 8

Q1. The mean free path of air molecules at STP is \(2.9 \times 10^{-7} \, \text{m}\). If temperature doubles at constant pressure, what is the new mean free path?

Q2. At what temperature is the rms speed of oxygen molecules 964 m/s? (Molecular mass of \(O_2 = 32 \, \text{u}\), \(k_B = 1.38 \times 10^{-23} \, \text{J K}^{-1}\))

Q3. A mixture of 0.1 moles of helium and 0.9 moles of oxygen is at 500 K in a 30-litre container. What is the total pressure? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q4. What is the average translational kinetic energy of a hydrogen molecule at 700 K? (\(k_B = 1.38 \times 10^{-23} \, \text{J K}^{-1}\))

Q5. A gas at 1.5 atm and 300 K has a volume of 12 litres. If the temperature increases to 450 K at constant pressure, what is the new volume?

Q6. The mean free path of a gas molecule is \(1.5 \times 10^{-6} \, \text{m}\) at 0.2 atm. What will it be at 0.8 atm if temperature remains constant?

Q7. What is the total internal energy of 0.4 moles of a diatomic gas at 400 K with no vibrational modes? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q8. A gas at 1.5 atm and 300 K has a density of \(1.2 \, \text{kg m}^{-3}\). What is its molecular mass? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\), 1 atm = \(1.01 \times 10^5 \, \text{Pa}\))

Q9. A gas has a \(C_v\) of \(20.8 \, \text{J mol}^{-1} \text{K}^{-1}\). What is the ratio of specific heats (\(\gamma\))? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q10. The mean free path of a gas molecule is \(1.0 \times 10^{-6} \, \text{m}\) at 0.5 atm. What will it be at 2 atm if temperature remains constant?