Choose the correct answers

Physics : Kinetic Theory of Gases

Multiple choice questions

1. A particle of mass m is moving with speed u in a direction which makes 60° with respect to x axis. It undergoes elastic collision with the wall. What is the change in momentum in x and y direction?

(a) ∆px = −mu, ∆py = 0

(b) ∆px = −2mu, ∆py = 0

(c) ∆px = 0, ∆py = mu

(d) ∆px = mu, ∆py = 0

Answer : (a) Δp_x = −mu, Δp_y = 0

Solution :

Δp_x = 2mucos60

= − 2mu × ½ = (−2 mu ) / 2 = −mu

ΔP_y = 0 

2. A sample of ideal gas is at equilibrium. Which of the following quantity is zero?

(a) rms speed

(b) average speed

(c) average velocity

(d) most probable speed

Answer : (c) average velocity

3. An ideal gas is maintained at constant pressure. If the temperature of an ideal gas increases from 100K to 1000K then the rms speed of the gas molecules

(a) increases by 5 times

(b) increases by 10 times

(c) remains same

(d) increases by 7 times

Answer : (b) increases by 10 times

Solutions :

V_rms = √[3RT / m]

V_rms ∝√T

V_rms ∝√100

V´_rms ∝√1000

V´_rms / V_rms = √(1000 / 100)

V'_rms = √10 V_rms

4.   Two identically sized rooms A and B are connected by an open door. If the room A is air conditioned such that its temperature is 4° lesser than room B, which room has more air in it?

(a) Room A

(b) Room B

(c) Both room has same air

(d) Cannot be determined

Answer : (a) Room A

5. The average translational kinetic energy of gas molecules depends on

(a) number of moles and T

(b) only on T

(c) P and T

(d) P only

Answer: (a) number of moles and T

Solution :

(K = μR)

K.E = 3/2 KT

= 3/2 μRT

6.   If the internal energy of an ideal gas U and volume V are doubled then the pressure

(a) doubles

(b) remains same

(c) halves

(d) quadruples C

Solution : P = (2/3) (u/v)

P' = (2/3) × (2u / 2v )

P' = P 

7.   The ratio γ = Cp/Cv for a gas mixture consisting of 8 g of helium and 16 g of oxygen is

(a) 23/15

(b) 15/23

(c) 27/11

(d) 17/27

Answer : (c) 27/17

Solution:

γ = C_P / C_V = ( 5μ₁ + 7μ₂ ) / ( 3μ₁ + 5μ₂)

= [ 5 × 2 + (7 × 1/2 ) ] / [ 3 × 2 + (5 × 1/2) ]

= [ 10 + 7/2 ] / [ 6 + 5/2 ] = [27/2] / [17/2]

γ = 27 / 17

8. A container has one mole of monoatomic ideal gas. Each molecule has f degrees of freedom. What is the ratio of γ = Cp/Cv

(a) f

(b) f/2

(c) f / [ f +2]

(d) [f +2] / f

Answer: (d) (f + 2) / f

Solution:

γ = C_P / C_V

C_V = 3/2 R = f /2 R

C_P = 5/2 R = [ (f + 2) / 2 ] R

γ = { R [ (f + 2) / 2 ] } / { R (f / 2) }

γ = ( f + 2) / f

9.  If the temperature and pressure of a gas is doubled the mean free path of the gas molecules

(a) remains same

(b) doubled

(c) tripled

(d) quadrapoled

Answer : (a) remains same

Solution:

λ = kT / [√2 πd²P]

λ´ = [ k (2T) ] / [√2 πd² (2P) ]

λ = λ´

10. Which of the following shows the correct relationship between the pressure and density of an ideal gas at constant temperature?

Answer: (d) 

Solution:

P = 1/3 ρV²,

  ρ ∝P

11. A sample of gas consists of μ1 moles of monoatomic molecules, μ2 moles of diatomic molecules and μ3 moles of linear triatomic molecules. The gas is kept at high temperature. What is the total number of degrees of freedom?

(a) [3μ1 + 7( μ2 + μ3)] NA

(b) [3μ1 + 7 μ2 + 6μ3] NA

(c) [7μ1 + 3( μ2 + μ3)] NA

(d) [3μ1 + 6( μ2 + μ3)] NA

Answer : (a) [3μ₁ + 7( μ₂ + μ₃)] N_A

Solution:

For a monoatomic molecule, f = 3

For a diatomic molecule, f = 7 (H.T)

For a triatomic molecule, f = 7 (L.T)

Total degrees of freedom, f = [3μ₁ + 7( μ₂ + μ₃)] N_A

12. If sP and sV denote the specific heats of nitrogen gas per unit mass at constant pressure and constant volume respectively, then

(a) sP - sV = 28R

(b) sP - sV = R/28

(c) sP - sV  = R/14

(d) sP - sV = R

Answer: (b) s_P − s_v = R/28

Solution:

V = μC_V ΔT;

V= μC_P ΔT

mS_V = μC_V

(m/μ) S_V = C_V

mS_V = C_V

For N₂ → M = 28

28 S_V = C_V

28S_P = C_P

C_P – C_V = R

28 (S_P − S_V) = R

S_P − S_V = R / 28

13.  Which of the following gases will have least rms speed at a given temperature?

(a) Hydrogen

(b) Nitrogen

(c) Oxygen

(d) Carbon dioxide

Answer : (d) Carbon dioxide

Solution:

V_rms = √[3RT / M]

V_rms ∝ 1 / √M

14.  For a given gas molecule at a fixed temperature,  the  area  under  the Maxwell-Boltzmann distribution curve is equal to

(a) PV/kT

(b) kT/PV

(c) P/NkT

(d) PV

Answer: (a) PV / kT

Solution:

PV = NkT

N = PV / kT

15. The following graph represents the pressure versus number density for ideal gas at two different temperatures T1 and T2. The graph implies

(a) T1 = T2

(b) T1 > T2

(c) T1 < T2

d) Cannot be determined

Answer : (b) T₁ > T₂

Solution:

P = 2/3 U,

U∝T,

P∝T

Answers:           

1) a  2) c     3) b   4) a

5) a  6) b     7) c    8) d

9) a  10) d   11) a  12) b

13) d 14) a  15) b