JEE Physics Kinetic Theory MCQs Set 17

Multiple Choice Questions (MCQs) for JEE Physics: Kinetic Theory of gases

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Multiple Choice Questions

Question 1: The pressure exerted by a perfect gas is equal to
(a) mean kinetic energy per unit volume
(b) half of the mean kinetic energy per unit volume
(c) two-thirds of the mean kinetic energy per unit volume
(d) one-third of the mean kinetic energy per unit volume
Show Answer & Explanation

Answer: (c) two-thirds of the mean kinetic energy per unit volume

Explanation:
1. P = (1/3)ρ⟨v²⟩ and the translational kinetic energy per unit volume is (1/2)ρ⟨v²⟩.
2. So P = (2/3) × (energy per unit volume).

Teacher's Note:
1. P = (2/3)E/V.

Question 2: In the kinetic theory of gases, one assumes that the collisions between the molecules are
(a) perfectly elastic
(b) perfectly inelastic
(c) partly inelastic
(d) may be perfectly elastic or perfectly inelastic depending on the nature of the gas
Show Answer & Explanation

Answer: (a) perfectly elastic

Explanation:
1. Kinetic energy must be conserved in the collisions for the temperature to stay constant.
2. So the collisions are perfectly elastic.

Teacher's Note:
1. Elastic collisions.

Question 3: The ratio of the molar specific heat at constant pressure C_P to that at constant volume C_V for a monoatomic gas is
(a) \( \frac{C_P}{C_V}=\frac35 \)
(b) \( \frac{C_P}{C_V}=\frac53 \)
(c) \( \frac{C_P}{C_V}=\frac79 \)
(d) \( \frac{C_P}{C_V}=\frac97 \)
Show Answer & Explanation

Answer: (b) \( \frac{C_P}{C_V}=\frac53 \)

Explanation:
1. For a monoatomic gas C_V = 3R/2 and C_P = 5R/2.
2. The ratio is 5/3.

Teacher's Note:
1. γ = 5/3.

Question 4: What will be the ratio of the number of molecules of a monoatomic and a diatomic gas in a vessel if the ratio of their partial pressures is 5 : 3?
(a) 5 : 1
(b) 3 : 1
(c) 5 : 3
(d) 3 : 5
Show Answer & Explanation

Answer: (c) 5 : 3

Explanation:
1. Partial pressure P = nkT/V for each gas at the same T and V.
2. So the number ratio equals the pressure ratio, 5 : 3.

Teacher's Note:
1. P ∝ N at fixed T and V.

Question 5: If a gas is heated at constant pressure, its isothermal compressibility
(a) remains constant
(b) increases linearly with temperature
(c) decreases linearly with temperature
(d) decreases inversely with temperature
Show Answer & Explanation

Answer: (a) remains constant

Explanation:
1. For an ideal gas the isothermal compressibility is 1/P.
2. At constant pressure it stays constant.

Teacher's Note:
1. K = 1/P.

Question 6: The temperature at which oxygen molecules have the same root mean square speed as hydrogen molecules at 300 K is
(a) 600 K
(b) 2400 K
(c) 4800 K
(d) 300 K
Show Answer & Explanation

Answer: (c) 4800 K

Explanation:
1. v_rms ∝ √(T/M), so T ∝ M for equal speeds.
2. T = 300 × (32/2) = 4800 K.

Teacher's Note:
1. T/M must be equal.

Question 7: The absolute temperature of a gas determines
(a) the average momentum of the molecule
(b) the velocity of sound in the gas
(c) the number of molecules in the gas
(d) the mean square velocity of the molecules
Show Answer & Explanation

Answer: (d) the mean square velocity of the molecules

Explanation:
1. Since (1/2)m⟨v²⟩ = (3/2)kT, the mean square velocity is fixed by T and the molecular mass.
2. The other quantities are not determined by T alone.

Teacher's Note:
1. ⟨v²⟩ = 3kT/m.

Question 8:
Assertion (A): Each vibrational mode gives two degrees of freedom.
Reason (R): By the law of equipartition of energy, the energy for each degree of freedom in thermal equilibrium is 2k_BT.
(a) Both A and R are true, and R is the correct explanation of A.
(b) Both A and R are true, but R is not the correct explanation of A.
(c) A is true, but R is false.
(d) A is false, but R is true.
Show Answer & Explanation

Answer: (c) A is true, but R is false.

Explanation:
1. A vibrational mode has kinetic and potential terms, so it counts as two degrees of freedom.
2. The energy per degree of freedom is (1/2)k_BT, so the reason is false.

Teacher's Note:
1. kT/2 per degree.

Question 9: If E is the translational kinetic energy of a gas of volume V and pressure P, then
(a) \( PV=E \)
(b) \( PV=\frac32E \)
(c) \( PV=3E \)
(d) \( PV=\frac23E \)
Show Answer & Explanation

Answer: (d) \( PV=\frac23E \)

Explanation:
1. P = (2/3)(E/V) where E is the total translational kinetic energy.
2. So PV = (2/3)E.

Teacher's Note:
1. PV = 2E/3.

Question 10: If a gas has n degrees of freedom, the ratio of the specific heats γ of the gas is
(a) \( \frac{1+n}{2} \)
(b) \( 1+\frac n2 \)
(c) \( 1+\frac1n \)
(d) \( 1+\frac2n \)
Show Answer & Explanation

Answer: (d) \( 1+\frac2n \)

Explanation:
1. C_V = nR/2 and C_P = C_V + R.
2. γ = 1 + R/C_V = 1 + 2/n.

Teacher's Note:
1. γ = 1 + 2/f.

Question 11: The temperature at which the average translational kinetic energy of a gas molecule equals the energy gained by an electron accelerated from rest through a potential difference of 1 volt is nearly
(a) 4.6 × 10³ K
(b) 11.6 × 10³ K
(c) 23.2 × 10³ K
(d) 7.7 × 10³ K
Show Answer & Explanation

Answer: (d) 7.7 × 10³ K

Explanation:
1. (3/2)kT = 1.6 × 10⁻¹⁹ J.
2. T = 2 × 1.6 × 10⁻¹⁹/(3 × 1.38 × 10⁻²³) ≈ 7.7 × 10³ K.

Teacher's Note:
1. Set (3/2)kT = 1 eV.

Question 12: The molar specific heats of an ideal gas at constant pressure and volume are C_P and C_V respectively. If γ = C_P/C_V and R is the universal gas constant, then C_V is equal to
(a) \( \frac{R}{\gamma-1} \)
(b) \( \frac{\gamma-1}{R} \)
(c) \( \gamma R \)
(d) \( \frac{1+\gamma}{1-\gamma} \)
Show Answer & Explanation

Answer: (a) \( \frac{R}{\gamma-1} \)

Explanation:
1. C_P = γC_V and C_P − C_V = R.
2. So (γ − 1)C_V = R and C_V = R/(γ − 1).

Teacher's Note:
1. Combine γ and Mayer's relation.

Question 13: For Boyle's law to hold, the gas should be
(a) perfect and of constant mass and temperature
(b) real and of constant mass and temperature
(c) perfect and of constant temperature but variable mass
(d) real and of constant temperature but variable mass
Show Answer & Explanation

Answer: (a) perfect and of constant mass and temperature

Explanation:
1. Boyle's law is derived for an ideal gas.
2. PV = nRT gives PV = constant only if n and T are fixed.

Teacher's Note:
1. Fixed n and T for an ideal gas.

Question 14:
Assertion (A): The total translational kinetic energy of all the molecules of a given mass of an ideal gas is 1.5 times the product of its pressure and its volume.
Reason (R): The molecules of a gas collide with each other and the velocities of the molecules change due to the collision.
(a) Both A and R are true, and R is the correct explanation of A.
(b) Both A and R are true, but R is not the correct explanation of A.
(c) A is true, but R is false.
(d) A is false, but R is true.
Show Answer & Explanation

Answer: (b) Both A and R are true, but R is not the correct explanation of A.

Explanation:
1. Total translational KE = (3/2)PV, so the assertion is true.
2. The reason is a true statement but it does not explain the relation.

Teacher's Note:
1. KE = 3PV/2.

Question 15: According to the kinetic theory of gases, the pressure exerted by a gas on the wall of the container is measured as
(a) rate of change of momentum imparted to the walls per second per unit area
(b) momentum imparted to the walls per unit area
(c) change of momentum imparted to the walls per unit volume
(d) change in momentum per unit volume
Show Answer & Explanation

Answer: (a) rate of change of momentum imparted to the walls per second per unit area

Explanation:
1. Pressure is force per unit area.
2. Force equals the rate of change of momentum, so pressure is the momentum imparted per second per unit area.

Teacher's Note:
1. Pressure = force per area = rate of momentum transfer per area.

Question 16: The root mean square speed of a group of n gas molecules having speeds v₁, v₂, v₃, …, vₙ is
(a) \( \frac1n\sqrt{(v_1+v_2+\cdots+v_n)^2} \)
(b) \( \frac1n\sqrt{v_1^2+v_2^2+\cdots+v_n^2} \)
(c) \( \sqrt{\frac1n\left(v_1^2+v_2^2+\cdots+v_n^2\right)} \)
(d) \( \sqrt{\frac{(v_1+v_2+\cdots+v_n)^2}{n}} \)
Show Answer & Explanation

Answer: (c) \( \sqrt{\frac1n\left(v_1^2+v_2^2+\cdots+v_n^2\right)} \)

Explanation:
1. The rms speed is the square root of the mean of the squares of the speeds.
2. That is √[(v₁² + v₂² + … + vₙ²)/n].

Teacher's Note:
1. Root of the mean of squares.

Question 17: In a mixture of gases at a fixed temperature
(a) a heavier molecule has a higher average speed
(b) a lighter molecule has a lower average speed
(c) a heavier molecule has a lower average speed
(d) none of these
Show Answer & Explanation

Answer: (c) a heavier molecule has a lower average speed

Explanation:
1. At a given temperature all molecules have the same average kinetic energy (1/2)mv².
2. A larger mass therefore means a smaller average speed.

Teacher's Note:
1. v ∝ 1/√m at a fixed T.

Question 18: When the temperature is constant, the pressure of a given mass of gas varies inversely with volume. This is the statement of
(a) Boyle's law
(b) Charles' law
(c) Avogadro's law
(d) none of these
Show Answer & Explanation

Answer: (a) Boyle's law

Explanation:
1. PV = constant at constant temperature is Boyle's law.

Teacher's Note:
1. Boyle's law is isothermal.

Question 19: The pressure exerted on the walls of a container by a gas is due to the fact that the gas molecules are
(a) losing their kinetic energy
(b) sticking to the walls
(c) changing their momenta due to collision with the walls
(d) getting accelerated towards the wall
Show Answer & Explanation

Answer: (c) changing their momenta due to collision with the walls

Explanation:
1. Each molecule rebounds from the wall and its momentum changes.
2. The average rate of this momentum change over all molecules is the force on the wall.

Teacher's Note:
1. Collisions produce pressure.

Question 20: Two vessels separately contain two ideal gases A and B at the same temperature. The pressure of A is twice that of B. Under such conditions the density of A is 1.5 times the density of B. The ratio of the molecular weight of A to that of B is
(a) \( \frac34 \)
(b) 2
(c) \( \frac12 \)
(d) \( \frac23 \)
Show Answer & Explanation

Answer: (a) \( \frac34 \)

Explanation:
1. M = ρRT/P at the same T.
2. M_A/M_B = (ρ_A/ρ_B)(P_B/P_A) = 1.5/2 = 3/4.

Teacher's Note:
1. M = ρRT/P.

Question 21:
Assertion (A): For an ideal gas at constant temperature, the product of pressure and volume is constant.
Reason (R): The mean square velocity of the molecules is inversely proportional to mass.
(a) Both A and R are true, and R is the correct explanation of A.
(b) Both A and R are true, but R is not the correct explanation of A.
(c) A is true, but R is false.
(d) A is false, but R is true.
Show Answer & Explanation

Answer: (b) Both A and R are true, but R is not the correct explanation of A.

Explanation:
1. PV = nRT is constant at constant T.
2. The mean square speed ∝ 1/m is true at a given T, but it is not the reason for the constancy of PV.

Teacher's Note:
1. Boyle's law.

Question 22: The temperature of a gas is produced by
(a) the potential energy of its molecules
(b) the kinetic energy of its molecules
(c) the attractive force between its molecules
(d) the repulsive force between its molecules
Show Answer & Explanation

Answer: (b) the kinetic energy of its molecules

Explanation:
1. Temperature is a measure of the random kinetic energy of the molecules.
2. Potential energy plays no role for an ideal gas.

Teacher's Note:
1. Temperature comes from molecular motion.

Question 23: At a given temperature, which of the following gases possesses the maximum r.m.s. velocity?
(a) hydrogen
(b) oxygen
(c) nitrogen
(d) carbon dioxide
Show Answer & Explanation

Answer: (a) hydrogen

Explanation:
1. v_rms = √(3RT/M) is largest for the smallest molar mass.
2. Hydrogen has the smallest M.

Teacher's Note:
1. v_rms ∝ 1/√M.

Question 24:
Assertion (A): If a gas container in motion is suddenly stopped, the temperature of the gas rises.
Reason (R): The kinetic energy of ordered mechanical motion is converted into the kinetic energy of random motion of the gas molecules.
(a) Both A and R are true, and R is the correct explanation of A.
(b) Both A and R are true, but R is not the correct explanation of A.
(c) A is true, but R is false.
(d) A is false, but R is true.
Show Answer & Explanation

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. On stopping, the ordered kinetic energy of the gas becomes random molecular energy.
2. This raises the temperature, as the reason says.

Teacher's Note:
1. Ordered motion turns into random motion.

Question 25: A gas is enclosed in a cube of side l. What is the change in momentum of a molecule if it suffers an elastic collision with the plane wall parallel to the yz-plane and rebounds with the same speed? (vₓ, v_y, v_z are the initial velocity components.)
(a) \( mv_x \)
(b) zero
(c) \( -mv_x \)
(d) \( -2mv_x \)
Show Answer & Explanation

Answer: (d) \( -2mv_x \)

Explanation:
1. Only the x-component of velocity reverses.
2. The change in momentum is −mvₓ − mvₓ = −2mvₓ.

Teacher's Note:
1. Δp = −2mvₓ.

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