NEET Behaviour of Perfect Gas and Kinetic Theory — Set 11

Question 1

What is the average translational kinetic energy of a nitrogen molecule at 450 K? ($k_B = 1.38 \times 10^{-23} \, \text{J K}^{-1}$)

Accepted Answer

Average translational KE = $\frac{3}{2} k_B T$. $\frac{3}{2} \times 1.38 \times 10^{-23} \times 450 = 9.315 \times 10^{-21} \, \text{J}$.

Question 2

What is the pressure of 0.4 moles of an ideal gas in a 4-litre container at 327°C? ($R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}$)

Accepted Answer

$PV = \mu R T$, $P = \frac{\mu R T}{V}$. $T = 327 + 273 = 600 \, \text{K}$, $V = 4 \times 10^{-3} \, \text{m}^3$. $P = \frac{0.4 \times 8.31 \times 600}{4 \times 10^{-3}} = 4.986 \times 10^5 \, \text{Pa} \approx 5.0 \, \text{atm} (1 \, \text{atm} \approx 10^5 \, \text{Pa})$.

Question 3

The mean free path of a gas is $1.8 \times 10^{-7} \, \text{m}$ with a number density of $3.0 \times 10^{25} \, \text{m}^{-3}$. What is the molecular diameter?

Accepted Answer

$l = \frac{1}{\sqrt{2} n \pi d^2}$, $d^2 = \frac{1}{\sqrt{2} n \pi l}$. $d^2 = \frac{1}{1.414 \times 3.0 \times 10^{25} \times 3.14 \times 1.8 \times 10^{-7}} = \frac{1}{2.4 \times 10^{-19}} \approx 4.17 \times 10^{-20}$. $d = \sqrt{4.17 \times 10^{-20}} \approx 2.04 \times 10^{-10} \, \text{m}$.

NEET Physics: Behaviour of Perfect Gas and Kinetic Theory — Practice Set 11

Q1. What is the average translational kinetic energy of a nitrogen molecule at 450 K? (\(k_B = 1.38 \times 10^{-23} \, \text{J K}^{-1}\))

Q2. What is the pressure of 0.4 moles of an ideal gas in a 4-litre container at 327°C? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q3. The mean free path of a gas is \(1.8 \times 10^{-7} \, \text{m}\) with a number density of \(3.0 \times 10^{25} \, \text{m}^{-3}\). What is the molecular diameter?

Q4. At what temperature is the rms speed of hydrogen molecules 2000 m/s? (Molecular mass of \(H_2 = 2 \, \text{u}\), \(k_B = 1.38 \times 10^{-23} \, \text{J K}^{-1}\))

Q5. The rms speed of nitrogen molecules is 516 m/s at 300 K. What is the rms speed of oxygen molecules at the same temperature? (Molecular mass: \(N_2 = 28 \, \text{u}\), \(O_2 = 32 \, \text{u}\))

Q6. A gas has a density of \(0.8 \, \text{kg m}^{-3}\) at 1 atm and 300 K. What is its molecular mass? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\), 1 atm = \(1.01 \times 10^5 \, \text{Pa}\))

Q7. What is the volume of 0.6 moles of an ideal gas at 3 atm and 527°C? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q8. A solid has a molar specific heat capacity of \(24.5 \, \text{J mol}^{-1} \text{K}^{-1}\). How many degrees of freedom per atom does it have? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q9. The rms speed of helium molecules is 1370 m/s at 300 K. What is the rms speed of argon molecules at the same temperature? (Atomic mass: He = 4 u, Ar = 39.9 u)

Q10. A gas at 3 atm and 600 K has a volume of 15 litres. If the pressure decreases to 1.5 atm at constant temperature, what is the new volume?