NEET Behaviour of Perfect Gas and Kinetic Theory — Set 5

Question 1

The mean free path of a gas is $1.2 \times 10^{-7} \, \text{m}$ with a number density of $4.0 \times 10^{25} \, \text{m}^{-3}$. What is the molecular diameter?

Accepted Answer

$l = \frac{1}{\sqrt{2} n \pi d^2}$, $d^2 = \frac{1}{\sqrt{2} n \pi l}$. $d^2 = \frac{1}{1.414 \times 4.0 \times 10^{25} \times 3.14 \times 1.2 \times 10^{-7}} = \frac{1}{2.13 \times 10^{-19}} \approx 4.69 \times 10^{-20}$. $d = \sqrt{4.69 \times 10^{-20}} \approx 2.17 \times 10^{-10} \, \text{m}$.

Question 2

A gas has a $C_v$ of $27.0 \, \text{J mol}^{-1} \text{K}^{-1}$. What is its $C_p$? ($R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}$)

Accepted Answer

$C_p = C_v + R$. $C_p = 27.0 + 8.31 = 35.31 \, \text{J mol}^{-1} \text{K}^{-1} \approx 35.3 \, \text{J mol}^{-1} \text{K}^{-1}$.

Question 3

The rms speed of a gas is 550 m/s at 275 K. At what temperature will the rms speed be 1100 m/s?

Accepted Answer

$v_{\text{rms}} \propto \sqrt{T}$, $\frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}}$. $\frac{1100}{550} = \sqrt{\frac{T_2}{275}}$, $2 = \sqrt{\frac{T_2}{275}}$. Square both sides: $4 = \frac{T_2}{275}$, $T_2 = 1100 \, \text{K}$.

NEET Physics: Behaviour of Perfect Gas and Kinetic Theory — Practice Set 5

Q1. The mean free path of a gas is \(1.2 \times 10^{-7} \, \text{m}\) with a number density of \(4.0 \times 10^{25} \, \text{m}^{-3}\). What is the molecular diameter?

Q2. A gas has a \(C_v\) of \(27.0 \, \text{J mol}^{-1} \text{K}^{-1}\). What is its \(C_p\)? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q3. The rms speed of a gas is 550 m/s at 275 K. At what temperature will the rms speed be 1100 m/s?

Q4. A gas has a \(C_v\) of \(17.1 \, \text{J mol}^{-1} \text{K}^{-1}\). What is its \(C_p\)? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q5. A gas has a volume of 22.4 litres at STP. How many moles are present if the temperature is raised to 546 K at constant pressure?

Q6. What is the temperature at which the rms speed of nitrogen molecules is 1032 m/s? (Molecular mass of \(N_2 = 28 \, \text{u}\), \(k_B = 1.38 \times 10^{-23} \, \text{J K}^{-1}\))

Q7. What is the collision frequency of a gas molecule with a mean free path of \(1.2 \times 10^{-7} \, \text{m}\) and average speed of 480 m/s?

Q8. A gas has a molar specific heat at constant volume of \(20.8 \, \text{J mol}^{-1} \text{K}^{-1}\). What is its \(C_p\)? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q9. What is the average translational kinetic energy of a nitrogen molecule (\(N_2\)) at 300 K? (\(k_B = 1.38 \times 10^{-23} \, \text{J K}^{-1}\))

Q10. A gas mixture has equal masses of helium and oxygen. What is the ratio of their number of molecules?