NEET Behaviour of Perfect Gas and Kinetic Theory — Set 6

Question 1

The number density of a gas is $2.5 \times 10^{25} \, \text{m}^{-3}$. If the molecular diameter is $2 \times 10^{-10} \, \text{m}$, what is the mean free path?

Accepted Answer

$l = \frac{1}{\sqrt{2} n \pi d^2}$. $n = 2.5 \times 10^{25}$, $d = 2 \times 10^{-10}$, $\pi \approx 3.14$. $l = \frac{1}{1.414 \times 2.5 \times 10^{25} \times 3.14 \times (2 \times 10^{-10})^2} = \frac{1}{4.44 \times 10^{17}} \approx 2.25 \times 10^{-7} \, \text{m} $.

Question 2

What is the total internal energy of 0.5 moles of a monatomic gas at 600 K? ($R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}$)

Accepted Answer

Monatomic gas: $U = \frac{3}{2} \mu R T$. $U = \frac{3}{2} \times 0.5 \times 8.31 \times 600 = 3741.75 \, \text{J} \approx 3.74 \, \text{kJ}$.

Question 3

A gas mixture contains 4 g of helium and 32 g of oxygen. What is the total pressure if the volume is 22.4 litres at 273 K? ($R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}$)

Accepted Answer

$\mu_{\text{He}} = \frac{4}{4} = 1 \, \text{mol}$, $\mu_{\text{O}_2} = \frac{32}{32} = 1 \, \text{mol}$. Total moles = $1 + 1 = 2 \, \text{mol}$, $V = 22.4 \times 10^{-3} \, \text{m}^3$. $P = \frac{\mu R T}{V} = \frac{2 \times 8.31 \times 273}{22.4 \times 10^{-3}} = 2.02 \times 10^5 \, \text{Pa} \approx 2 \, \text{atm} $.

NEET Physics: Behaviour of Perfect Gas and Kinetic Theory — Practice Set 6

Q1. The number density of a gas is \(2.5 \times 10^{25} \, \text{m}^{-3}\). If the molecular diameter is \(2 \times 10^{-10} \, \text{m}\), what is the mean free path?

Q2. What is the total internal energy of 0.5 moles of a monatomic gas at 600 K? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q3. A gas mixture contains 4 g of helium and 32 g of oxygen. What is the total pressure if the volume is 22.4 litres at 273 K? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q4. A solid has a molar specific heat capacity of \(24.9 \, \text{J mol}^{-1} \text{K}^{-1}\). What is the energy per degree of freedom per atom? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q5. What is the collision frequency of a gas molecule with a mean free path of \(1.0 \times 10^{-7} \, \text{m}\) and average speed of 400 m/s?

Q6. The rms speed of oxygen molecules at a certain temperature is 500 m/s. What is the rms speed of hydrogen molecules at the same temperature? (Molecular mass of \(O_2 = 32 \, \text{u}\), \(H_2 = 2 \, \text{u}\))

Q7. A gas has a \(C_v\) of \(20.4 \, \text{J mol}^{-1} \text{K}^{-1}\). What is its \(C_p\)? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q8. A gas mixture has equal masses of hydrogen and argon at 300 K. What is the ratio of their rms speeds? (Molecular mass: \(H_2 = 2 \, \text{u}\), Ar = 39.9 u)

Q9. A gas mixture contains 2 g of hydrogen and 32 g of oxygen. What is the ratio of their partial pressures?

Q10. A gas at 2 atm and 400 K has a volume of 8 litres. If the temperature decreases to 200 K at constant pressure, what is the new volume?