Kinetic Theory of Gases and ldeal Gas | Physics JEE previous year questions with Solutions

Q 1 :
If the collision frequency of hydrogen molecules in a closed chamber at 27°C is Z, then the collision frequency of the same system at 127°C is [2024]

$\frac{3}{4} Z$
$\frac{\sqrt{3}}{2} Z$
$\frac{4}{3} Z$
$\frac{2}{\sqrt{3}} Z$

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(D) Collision frequency $\propto \sqrt{T} \Rightarrow \frac{f_{2}}{f_{1}} = \sqrt{\frac{T_{2}}{T_{1}}}$

$\Rightarrow \frac{f_{2}}{f_{1}} = \sqrt{\frac{400}{300}} \Rightarrow f_{2} = \frac{2}{\sqrt{3}} f_{1} = \frac{2}{\sqrt{3}} Z$

$f_{2} = \sqrt{\frac{4}{3}} = f_{1} = \frac{2}{\sqrt{3}} Z$

Q 2 :
If $n$ is the number density and $d$ is the diameter of the molecule, then the average distance covered by a molecule between two successive collisions (i.e. mean free path) is represented by [2024]

$\frac{1}{\sqrt{2} n {πd}^{2}}$
$\sqrt{2} n {πd}^{2}$
$\frac{1}{\sqrt{2 n {πd}^{2}}}$
$\frac{1}{\sqrt{2} n^{2} π^{2} d^{2}}$

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(A) The mean free path, $λ = \frac{1}{\sqrt{2} n {πd}^{2}}$

$n =$ number of molecules per unit volume,

and $d =$ diameter of the molecule

Q 3 :
A sample contains mixture of helium and oxygen gas. The ratio of root mean square speed of helium and oxygen in the sample, is [2024]

$\frac{1}{32}$
$\frac{1}{2 \sqrt{2}}$
$\frac{1}{4}$
$\frac{2 \sqrt{2}}{1}$

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(D) $V_{r m s} = \sqrt{\frac{3 R T}{M}}$

$\frac{V_{H e}}{V_{O_{2}}} = \sqrt{\frac{M_{O_{2}}}{M_{H e}}} = \sqrt{\frac{32}{4}} = \sqrt{8} = 2 \sqrt{2}$

Q 4 :
Given below are two statements:

Statement (I): The mean free path of gas molecules is inversely proportional to square of molecular diameter.

Statement (II): Average kinetic energy of gas molecules is directly proportional to absolute temperature of gas.

In the light of the above statements, choose the correct answer from the options given below: [2024]

Statement I is false but Statement II is true
Both Statement I and Statement II are true
Statement I is true but Statement II is false
Both Statement I and Statement II are false

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(B) Statement 1: Mean free path

$λ = \frac{1}{\sqrt{2} {πd}^{2} n}$ using $d = 2 r$

$λ = \frac{1}{4 \sqrt{2} π r^{2} n} \Rightarrow λ \propto \frac{1}{r^{2}}$

Statement 2:

$E = \frac{3}{2} k T \Rightarrow E \propto T$

Q 5 :
The temperature of a gas is – 78°C and the average translational kinetic energy of its molecules is K. The temperature at which the average translational kinetic energy of the molecules of the same gas becomes 2K is [2024]

– 39°C
117°C
127°C
– 78°C

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(B) Average translational kinetic energy

$E = \frac{3}{2} K_{B} T$

At – 78°C = 273 – 78 = 195 K

$K = \frac{3}{2} K_{B} \times (195 K)$ ...(i)

Now when kinetic energy becomes 2K, the temperature is T K.

$∴$ $2 K = \frac{3}{2} K_{B} \times (T K)$ ...(ii)

From equations (i) and (ii)

$\frac{K}{2 K} = \frac{195}{T}$

$\Rightarrow$ T = 390 K

$∴$ in °C, 390 – 273 = 117°C

Q 6 :
Two vessels A and B are of the same size and are at same temperature. A contains 1 g of hydrogen and B contains 1 g of oxygen. $P_{A}$ and $P_{B}$ are the pressures of the gases in A and B respectively, then $P_{A} / P_{B}$ is [2024]

16
8
4
32

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(A) Ideal gas equation, $P_{A} V_{A} = n_{A} R T_{A}$

and $P_{B} V_{B} = n_{B} R T_{B}$

from (i) and (ii), $\frac{P_{A} V_{A}}{P_{B} V_{B}} = \frac{n_{A} R T_{A}}{n_{B} R T_{B}}$

Given $V_{A} = V_{B}$ and $T_{A} = T_{B}$

$\frac{P_{A}}{P_{B}} = \frac{n_{A}}{n_{B}} = \frac{1 / 2}{1 / 32} = 16$

Q 7 :
The temperature of a gas having $2.0 \times 10^{25}$ molecules per cubic meter at 1.38 atm is

(Given, $k = 1.38 \times 10^{- 23} J K^{- 1}$ ) [2024]

300 K
500 K
100 K
200 K

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(B) ldeal gas equation,

$P V = \frac{N}{N_{A}} R T$

N = Total no. of molecules

$P = \frac{N}{V} k T$

$1.38 \times 1.01 \times 10^{5} = 2 \times 10^{25} \times 1.38 \times 10^{- 23} \times T$

$\Rightarrow T = \frac{1.01 \times 10^{3}}{2} \approx 500 K$

Q 8 :
At what temperature the r.m.s. velocity of a hydrogen molecule equal to that of an oxygen molecule at 47°C ? [2024]

80 K
– 73 K
4 K
20 K

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(D) $V_{r m s} = \sqrt{\frac{3 R T}{M}} \Rightarrow {(V_{r m s})}_{H_{2}} = {(V_{r m s})}_{O_{2}}$

$\Rightarrow \frac{T_{1}}{M_{1}} = \frac{T_{2}}{M_{2}}$

$\Rightarrow T_{1} = \frac{M_{1}}{M_{2}} T_{2} = \frac{2}{32} (273 + 47)$

$= \frac{2}{32} \times 320 = 20 K$

Q 9 :
If the root mean square velocity of hydrogen molecule at a given temperature and pressure is 2 km/s, the root mean square velocity of oxygen at the same condition in km/s is [2024]

2.0
0.5
1.5
1.0

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(B) $\frac{{(V_{r m s})}_{H_{2}}}{{(V_{r m s})}_{O_{2}}} = \frac{\sqrt{\frac{3 R T}{M_{H_{2}}}}}{\sqrt{\frac{3 R T}{M_{O_{2}}}}}$

$\Rightarrow \frac{2}{{(V_{r m s})}_{O_{2}}} = \sqrt{\frac{M_{O_{2}}}{M_{H_{2}}}} = \sqrt{\frac{32}{2}}$

$\Rightarrow {(V_{r m s})}_{O_{2}} = 0.5$

Topic Question Set

Q 1 :
If the collision frequency of hydrogen molecules in a closed chamber at 27°C is Z, then the collision frequency of the same system at 127°C is [2024]

Q 2 :
If $n$ is the number density and $d$ is the diameter of the molecule, then the average distance covered by a molecule between two successive collisions (i.e. mean free path) is represented by [2024]

Q 3 :
A sample contains mixture of helium and oxygen gas. The ratio of root mean square speed of helium and oxygen in the sample, is [2024]

Q 5 :
The temperature of a gas is – 78°C and the average translational kinetic energy of its molecules is K. The temperature at which the average translational kinetic energy of the molecules of the same gas becomes 2K is [2024]

Q 6 :
Two vessels A and B are of the same size and are at same temperature. A contains 1 g of hydrogen and B contains 1 g of oxygen. $P_{A}$ and $P_{B}$ are the pressures of the gases in A and B respectively, then $P_{A} / P_{B}$ is [2024]

Q 7 :
The temperature of a gas having $2.0 \times 10^{25}$ molecules per cubic meter at 1.38 atm is

(Given, $k = 1.38 \times 10^{- 23} J K^{- 1}$ ) [2024]

Q 8 :
At what temperature the r.m.s. velocity of a hydrogen molecule equal to that of an oxygen molecule at 47°C ? [2024]

Q 9 :
If the root mean square velocity of hydrogen molecule at a given temperature and pressure is 2 km/s, the root mean square velocity of oxygen at the same condition in km/s is [2024]

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Topic Question Set

Q 1 : If the collision frequency of hydrogen molecules in a closed chamber at 27°C is Z, then the collision frequency of the same system at 127°C is [2024]

Q 2 : If n is the number density and d is the diameter of the molecule, then the average distance covered by a molecule between two successive collisions (i.e. mean free path) is represented by [2024]

Q 3 : A sample contains mixture of helium and oxygen gas. The ratio of root mean square speed of helium and oxygen in the sample, is [2024]

Q 5 : The temperature of a gas is – 78°C and the average translational kinetic energy of its molecules is K. The temperature at which the average translational kinetic energy of the molecules of the same gas becomes 2K is [2024]

Q 6 : Two vessels A and B are of the same size and are at same temperature. A contains 1 g of hydrogen and B contains 1 g of oxygen. PA and PB are the pressures of the gases in A and B respectively, then PA/PB is [2024]

Q 7 : The temperature of a gas having 2.0×1025 molecules per cubic meter at 1.38 atm is (Given, k=1.38×10-23JK-1) [2024]

Q 8 : At what temperature the r.m.s. velocity of a hydrogen molecule equal to that of an oxygen molecule at 47°C ? [2024]

Q 9 : If the root mean square velocity of hydrogen molecule at a given temperature and pressure is 2 km/s, the root mean square velocity of oxygen at the same condition in km/s is [2024]

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Q 1 :
If the collision frequency of hydrogen molecules in a closed chamber at 27°C is Z, then the collision frequency of the same system at 127°C is [2024]

Q 2 :
If $n$ is the number density and $d$ is the diameter of the molecule, then the average distance covered by a molecule between two successive collisions (i.e. mean free path) is represented by [2024]

Q 3 :
A sample contains mixture of helium and oxygen gas. The ratio of root mean square speed of helium and oxygen in the sample, is [2024]

Q 5 :
The temperature of a gas is – 78°C and the average translational kinetic energy of its molecules is K. The temperature at which the average translational kinetic energy of the molecules of the same gas becomes 2K is [2024]

Q 6 :
Two vessels A and B are of the same size and are at same temperature. A contains 1 g of hydrogen and B contains 1 g of oxygen. $P_{A}$ and $P_{B}$ are the pressures of the gases in A and B respectively, then $P_{A} / P_{B}$ is [2024]

Q 7 :
The temperature of a gas having $2.0 \times 10^{25}$ molecules per cubic meter at 1.38 atm is

(Given, $k = 1.38 \times 10^{- 23} J K^{- 1}$ ) [2024]

Q 8 :
At what temperature the r.m.s. velocity of a hydrogen molecule equal to that of an oxygen molecule at 47°C ? [2024]

Q 9 :
If the root mean square velocity of hydrogen molecule at a given temperature and pressure is 2 km/s, the root mean square velocity of oxygen at the same condition in km/s is [2024]