NEET Behaviour of Perfect Gas and Kinetic Theory — Set 18

Question 1

What is the temperature at which the rms speed of helium atoms is 1000 m/s? (Atomic mass of He = 4 u, $k_B = 1.38 \times 10^{-23} \, \text{J K}^{-1}$)

Accepted Answer

$v_{\text{rms}} = \sqrt{\frac{3k_B T}{m}}$, $m = \frac{4 \times 10^{-3}}{6.02 \times 10^{23}} = 6.64 \times 10^{-27} \, \text{kg}$. $1000^2 = \frac{3 \times 1.38 \times 10^{-23} \times T}{6.64 \times 10^{-27}}$, $T = \frac{10^6 \times 6.64 \times 10^{-27}}{4.14 \times 10^{-23}} \approx 1604 \, \text{K} $.

Question 2

A gas has a $C_p$ of $33.24 \, \text{J mol}^{-1} \text{K}^{-1}$. What is its $C_v$? ($R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}$)

Accepted Answer

$C_p - C_v = R$, $C_v = C_p - R$. $C_v = 33.24 - 8.31 = 24.93 \, \text{J mol}^{-1} \text{K}^{-1}$.

Question 3

The rms speed of a gas molecule is 400 m/s at 200 K. What is the rms speed at 800 K?

Accepted Answer

$v_{\text{rms}} \propto \sqrt{T}$, $\frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}}$. $\frac{v_2}{400} = \sqrt{\frac{800}{200}} = \sqrt{4} = 2$. $v_2 = 400 \times 2 = 800 \, \text{m/s}$.

NEET Physics: Behaviour of Perfect Gas and Kinetic Theory — Practice Set 18

Q1. What is the temperature at which the rms speed of helium atoms is 1000 m/s? (Atomic mass of He = 4 u, \(k_B = 1.38 \times 10^{-23} \, \text{J K}^{-1}\))

Q2. A gas has a \(C_p\) of \(33.24 \, \text{J mol}^{-1} \text{K}^{-1}\). What is its \(C_v\)? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q3. The rms speed of a gas molecule is 400 m/s at 200 K. What is the rms speed at 800 K?

Q4. What is the pressure exerted by 0.1 mole of an ideal gas in a 2-litre container at 127°C? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q5. How much heat is required to raise the temperature of 1 mole of argon gas by 20 K at constant volume? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q6. The mean free path of a gas is \(7 \times 10^{-7} \, \text{m}\) with a number density of \(1.5 \times 10^{25} \, \text{m}^{-3}\). What is the molecular diameter?

Q7. The rms speed of a gas is 400 m/s at 100 K. At what temperature will the rms speed be 800 m/s?

Q8. A gas mixture has equal numbers of hydrogen and oxygen molecules at 300 K. What is the ratio of their rms speeds? (Molecular mass: \(H_2 = 2 \, \text{u}\), \(O_2 = 32 \, \text{u}\))

Q9. The rms speed of helium molecules is 1370 m/s at 300 K. What is the rms speed of oxygen molecules at the same temperature? (Atomic mass: He = 4 u, \(O_2 = 32 \, \text{u}\))

Q10. The mean free path of a gas molecule is \(6 \times 10^{-7} \, \text{m}\) at 0.5 atm. What will it be at 1 atm if temperature remains constant?