NEET Behaviour of Perfect Gas and Kinetic Theory — Set 12

Question 1

At what temperature is the rms speed of oxygen molecules 600 m/s? (Molecular mass of $O_2 = 32 \, \text{u}$, $k_B = 1.38 \times 10^{-23} \, \text{J K}^{-1}$)

Accepted Answer

$v_{\text{rms}} = \sqrt{\frac{3k_B T}{m}}$, $m = \frac{32 \times 10^{-3}}{6.02 \times 10^{23}} = 5.32 \times 10^{-26} \, \text{kg}$. $600^2 = \frac{3 \times 1.38 \times 10^{-23} \times T}{5.32 \times 10^{-26}}$, $T = \frac{3.6 \times 10^5 \times 5.32 \times 10^{-26}}{4.14 \times 10^{-23}} \approx 463 \, \text{K}$.

Question 2

A container holds 0.5 moles of an ideal gas at 2 atm pressure and 27°C. What is the volume of the gas? ($R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}$)

Accepted Answer

Ideal gas equation: $PV = \mu R T$. $T = 27 + 273 = 300 \, \text{K}$, $P = 2 \times 1.01 \times 10^5 \, \text{Pa} = 2.02 \times 10^5 \, \text{Pa}$, $\mu = 0.5 \, \text{mol}$. $V = \frac{\mu R T}{P} = \frac{0.5 \times 8.31 \times 300}{2.02 \times 10^5} = 6.17 \times 10^{-3} \, \text{m}^3 = 6.17 \, \text{litres} $.

Question 3

A mixture of 1 mole of neon and 2 moles of argon 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 = $1 + 2 = 3$, $V = 30 \times 10^{-3} \, \text{m}^3$. $P = \frac{3 \times 8.31 \times 500}{30 \times 10^{-3}} = 4.155 \times 10^5 \, \text{Pa} \approx 4.16 \, \text{atm}$.

NEET Physics: Behaviour of Perfect Gas and Kinetic Theory — Practice Set 12

Q1. At what temperature is the rms speed of oxygen molecules 600 m/s? (Molecular mass of \(O_2 = 32 \, \text{u}\), \(k_B = 1.38 \times 10^{-23} \, \text{J K}^{-1}\))

Q2. A container holds 0.5 moles of an ideal gas at 2 atm pressure and 27°C. What is the volume of the gas? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q3. A mixture of 1 mole of neon and 2 moles of argon 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. A vessel contains helium gas at 27°C with a pressure of 2 atm. If the temperature increases to 127°C at constant volume, what is the final pressure?

Q5. What is the pressure of 0.25 moles of an ideal gas in a 5-litre container at 227°C? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q6. A gas has a \(C_v\) of \(12.5 \, \text{J mol}^{-1} \text{K}^{-1}\). What is the ratio of specific heats (\(\gamma\))? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q7. The rms speed of nitrogen molecules is 516 m/s at 300 K. What is the rms speed of argon molecules at the same temperature? (Molecular mass: \(N_2 = 28 \, \text{u}\), Ar = 39.9 u)

Q8. What is the total internal energy of 1.5 moles of a monatomic gas at 250 K? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q9. A gas occupies 28.0 litres at STP. How many moles are present? (Molar volume at STP = 22.4 litres)

Q10. The mean free path of a gas molecule is \(2 \times 10^{-7} \, \text{m}\) at a certain pressure. If the pressure is doubled, what is the new mean free path? (Assume constant temperature)