NEET Behaviour of Perfect Gas and Kinetic Theory — Set 1

Question 1

The mean free path of a gas molecule is $4 \times 10^{-7} \, \text{m}$ at 2 atm. What will it be at 4 atm if temperature remains constant?

Accepted Answer

$l \propto \frac{1}{n}$, $n \propto P$. If $P$ doubles, $n$ doubles, $l$ halves. New $l = \frac{4 \times 10^{-7}}{2} = 2 \times 10^{-7} \, \text{m}$.

Question 2

The mean free path of a gas is $9 \times 10^{-7} \, \text{m}$ with a number density of $1.0 \times 10^{25} \, \text{m}^{-3}$. What is the molecular diameter?

Accepted Answer

$l = \frac{1}{\sqrt{2} n \pi d^2}$, $d^2 = \frac{1}{\sqrt{2} n \pi l}$. $d^2 = \frac{1}{1.414 \times 1.0 \times 10^{25} \times 3.14 \times 9 \times 10^{-7}} = \frac{1}{4.0 \times 10^{-19}} = 2.5 \times 10^{-20}$. $d = \sqrt{2.5 \times 10^{-20}} \approx 1.58 \times 10^{-10} \, \text{m}$.

Question 3

A solid has a molar specific heat capacity of $25.5 \, \text{J mol}^{-1} \text{K}^{-1}$. How many degrees of freedom per atom does it have? ($R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}$)

Accepted Answer

$C = f \times \frac{R}{2}$, $25.5 = f \times \frac{8.31}{2}$. $f = \frac{25.5 \times 2}{8.31} \approx 6.14 \approx 6$.

NEET Physics: Behaviour of Perfect Gas and Kinetic Theory — Practice Set 1

Q1. The mean free path of a gas molecule is \(4 \times 10^{-7} \, \text{m}\) at 2 atm. What will it be at 4 atm if temperature remains constant?

Q2. The mean free path of a gas is \(9 \times 10^{-7} \, \text{m}\) with a number density of \(1.0 \times 10^{25} \, \text{m}^{-3}\). What is the molecular diameter?

Q3. A solid has a molar specific heat capacity of \(25.5 \, \text{J mol}^{-1} \text{K}^{-1}\). How many degrees of freedom per atom does it have? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q4. A gas has a \(C_v\) of \(24.93 \, \text{J mol}^{-1} \text{K}^{-1}\). What is the ratio of specific heats (\(\gamma\))? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q5. A gas at 1.5 atm and 300 K has a volume of 24 litres. If the temperature increases to 600 K at constant pressure, what is the new volume?

Q6. The rms speed of a gas is 300 m/s at 150 K. At what temperature will the rms speed be 600 m/s?

Q7. The mean free path of a gas is \(6.0 \times 10^{-7} \, \text{m}\) with a number density of \(2.0 \times 10^{25} \, \text{m}^{-3}\). What is the molecular diameter?

Q8. What is the total internal energy of 0.8 moles of a triatomic gas at 500 K with no vibrational modes? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q9. How much heat is required to raise the temperature of 0.25 moles of a monatomic gas by 40 K at constant volume? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\))

Q10. A gas at 1.5 atm and 300 K has a density of \(0.72 \, \text{kg m}^{-3}\). What is its molecular mass? (\(R = 8.31 \, \text{J mol}^{-1} \text{K}^{-1}\), 1 atm = \(1.01 \times 10^5 \, \text{Pa}\))