Physics Objective Questions and Answers: Kinetic Theory of gases
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Multiple Choice Questions
(a) Only (1) and (2)
(b) Only (2) and (3)
(c) Only (1) and (4)
(d) Only (3) and (4)
Show Answer & Explanation
Answer: (c) Only (1) and (4)
Explanation:
1. (1) true: \( v_{rms}=\sqrt{3kT/m} \). (2) false: temperature is a property of a collection of particles.
2. (3) false: 100 °C to 200 °C is 373 K to 473 K, not a doubling.
3. (4) true: an isothermal process changes P and V while U stays constant.
Teacher's Note:
1. Temperature in the kinetic energy formula must be in kelvin.
2. Internal energy of an ideal gas depends only on temperature.
(a) \( 3.75\times10^5 \) N/m²
(b) \( 4.125\times10^5 \) N/m²
(c) \( 5.0\times10^5 \) N/m²
(d) \( 2.06\times10^5 \) N/m²
Show Answer & Explanation
Answer: (b) \( 4.125\times10^5 \) N/m²
Explanation:
1. Initial moles: \( N_2=0.05 \), He = 0.1. After dissociation: \( N_2=0.035 \), N atoms = 0.03.
2. Total moles = 0.035 + 0.03 + 0.1 = 0.165.
3. \( P=\dfrac{nRT}{V}=\dfrac{0.165(25/3)(1500)}{5\times10^{-3}}=4.125\times10^5 \) N/m².
Teacher's Note:
1. Each dissociating molecule produces two atoms, so the total number of particles rises.
2. Pressure is proportional to the total number of particles.
(a) 426.3 K
(b) 456.3 K
(c) 436.3 K
(d) 446.3 K
Show Answer & Explanation
Answer: (a) 426.3 K
Explanation:
1. \( v_{rms}\propto\sqrt{T/M} \): \( \dfrac{T_{O}}{32}=\dfrac{373}{28} \).
2. \( T_O=373\times\dfrac{32}{28}=426.3 \) K.
Teacher's Note:
1. Heavier molecules need a higher temperature for the same rms speed.
2. Convert 100 °C to 373 K.
(a) 2
(b) 1.5
(c) 4
(d) 3
Show Answer & Explanation
Answer: (d) 3
Explanation:
1. Partial pressure of B = 0.5 atm, so \( n_A:n_B=1:0.5=2:1 \) (same V and T).
2. \( \dfrac{2/M_A}{3/M_B}=2 \), so \( \dfrac{M_B}{M_A}=3 \), giving n = 3.
Teacher's Note:
1. Pressure is proportional to the number of moles at fixed V and T.
2. Moles = mass / molecular weight.
(a) 2 : 1
(b) 1 : 1
(c) 1 : 2
(d) 1 : 4
Show Answer & Explanation
Answer: (c) 1 : 2
Explanation:
1. Mean speed \( \propto\sqrt{T/M} \), so equal speeds need \( T_H/M_H=T_{He}/M_{He} \).
2. \( T_H/T_{He}=M_H/M_{He}=2/4=1/2 \).
Teacher's Note:
1. The lighter gas must be colder for the same mean speed.
2. Use molar masses 2 g/mol for hydrogen and 4 g/mol for helium.
(a) 13 kg
(b) 5.0 kg
(c) 20 kg
(d) 7.0 kg
Show Answer & Explanation
Answer: (d) 7.0 kg
Explanation:
1. Initial pressure \( P_i=10^5+\dfrac{130}{\pi(0.02)^2}=2.03\times10^5 \) Pa.
2. At constant volume \( P_f=P_i\dfrac{373}{293} \), so \( \Delta P=P_i\dfrac{80}{293}=5.55\times10^4 \) Pa.
3. Added weight = \( \Delta P\cdot A=69.8 \) N, so the added mass ≈ 7.0 kg.
Teacher's Note:
1. With the volume fixed, extra weight raises the pressure in proportion to the temperature.
2. Added mass = ΔP × area / g.
(a) 34 °C
(b) 32 °C
(c) 30 °C
(d) 37 °C
Show Answer & Explanation
Answer: (b) 32 °C
Explanation:
1. Both are 0.5 mol of diatomic gas with the same \( C_v \).
2. Energy conservation: \( 0.5(310)+0.5(300)=1.0\,T \), so T = 305 K = 32 °C.
Teacher's Note:
1. Equal moles of gases with equal \( C_v \) give the simple mean of the temperatures.
2. Convert to kelvin before averaging.
(a) the rotational kinetic energy
(b) the total kinetic energy
(c) none of these
(d) the average translational kinetic energy
Show Answer & Explanation
Answer: (d) the average translational kinetic energy
Explanation:
1. \( PV=\tfrac13Nm\overline{v^2} \), so \( \tfrac32PV=N\cdot\tfrac12m\overline{v^2} \).
2. This is the translational kinetic energy of the gas molecules.
Teacher's Note:
1. Rotation and vibration are not included in the pressure formula.
2. For a monatomic gas this is also the total internal energy.
(a) 4V
(b) \( \sqrt{\dfrac{50}{3}}\,V \)
(c) \( \sqrt{\dfrac{25}{3}}\,V \)
(d) \( \dfrac{50}{3}V \)
Show Answer & Explanation
Answer: (b) \( \sqrt{\dfrac{50}{3}}\,V \)
Explanation:
1. \( v_{rms}=\sqrt{\dfrac{v_1^2+v_2^2+v_3^2}{3}} \).
2. \( =\sqrt{\dfrac{9V^2+16V^2+25V^2}{3}}=\sqrt{\dfrac{50}{3}}\,V\approx4.08V \).
Teacher's Note:
1. The rms speed is the square root of the mean of the squares, not the mean of the speeds.
2. The mean speed here would be exactly 4V, which differs from the rms value.
(a) 2812.5 cal
(b) 1406.25 cal
(c) 1000 cal
(d) 5625 cal
Show Answer & Explanation
Answer: (a) 2812.5 cal
Explanation:
1. n = 3.125 mol of monatomic gas: \( U=n\tfrac32RT \).
2. \( U=3.125\times1.5\times2\times300=2812.5 \) cal.
Teacher's Note:
1. For a monatomic gas \( C_v=\tfrac32R \).
2. R is 2 cal/mol K in this problem.
(a) \( \dfrac{\ln\eta}{\ln(\Delta V/V)} \)
(b) \( \eta\,\dfrac{V}{\Delta V} \)
(c) \( \dfrac{\ln(1+\Delta V/V)}{\ln\eta} \)
(d) \( \dfrac{\ln\eta}{\ln(1+\Delta V/V)} \)
Show Answer & Explanation
Answer: (d) \( \dfrac{\ln\eta}{\ln(1+\Delta V/V)} \)
Explanation:
1. After one stroke \( P_1(V+\Delta V)=P_0V \), so \( P_1=P_0\dfrac{V}{V+\Delta V} \).
2. After n strokes \( P_n=P_0\left(\dfrac{V}{V+\Delta V}\right)^n=\dfrac{P_0}{\eta} \).
3. \( n=\dfrac{\ln\eta}{\ln(1+\Delta V/V)} \).
Teacher's Note:
1. Each stroke multiplies the pressure by the same factor.
2. Take logarithms to solve for n.
(a) 0.32
(b) 0.45
(c) 3.16
(d) 2.24
Show Answer & Explanation
Answer: (c) 3.16
Explanation:
1. Both gases are at the same temperature, so \( v_{rms}\propto1/\sqrt M \).
2. Ratio \( =\sqrt{40/4}=\sqrt{10}=3.16 \).
Teacher's Note:
1. The number of moles of each gas does not matter.
2. The lighter atoms are faster.
(a) Only (3) and (4)
(b) Only (1) and (2)
(c) Only (1) and (3)
(d) Only (2) and (4)
Show Answer & Explanation
Answer: (a) Only (3) and (4)
Explanation:
1. In a collision the total kinetic energy of the pair is conserved (elastic collision).
2. So one molecule can gain only what the other loses: the energies cannot both rise or both fall.
Teacher's Note:
1. Energy is transferred between the colliding molecules.
2. On average the two gases reach the same mean kinetic energy.
(a) its potential energy decreases
(b) its kinetic energy increases and the molecules move apart
(c) the number of collisions per unit area with the walls increases
(d) the energy of each molecule increases
Show Answer & Explanation
Answer: (c) the number of collisions per unit area with the walls increases
Explanation:
1. At constant temperature the average kinetic energy and speed of a molecule do not change.
2. Fewer volume means more molecules per unit volume, so more collisions per unit area per second on the walls.
Teacher's Note:
1. Pressure is the rate of momentum transfer to the walls.
2. Ideal gas molecules have no potential energy.
(a) increases as the temperature rises
(b) decreases as the temperature rises
(c) is proportional to the temperature
(d) remains constant
Show Answer & Explanation
Answer: (d) remains constant
Explanation:
1. \( U=\dfrac f2nRT=\dfrac f2PV \).
2. P is atmospheric and V is the room volume, both fixed, so U is constant (the number of molecules changes as air leaves).
Teacher's Note:
1. As the room warms, some air escapes, which exactly compensates the higher temperature.
2. Use PV in place of nRT.
(a) Only (1) and (3)
(b) Only (3) and (4)
(c) Only (2) and (4)
(d) Only (1) and (2)
Show Answer & Explanation
Answer: (d) Only (1) and (2)
Explanation:
1. \( C_v=\dfrac{Q}{n\Delta T}=\dfrac{25}{1\times2}=12.5 \) J/mol K ≈ \( \tfrac32R \).
2. \( \tfrac32R \) corresponds to a monatomic gas: helium or argon.
Teacher's Note:
1. At constant volume all the heat goes into internal energy.
2. Diatomic gases have \( C_v\approx\tfrac52R \).
(a) \( N_2 \), Ne
(b) He, \( H_2 \)
(c) Ar, \( O_2 \)
(d) \( O_2 \), Ar
Show Answer & Explanation
Answer: (c) Ar, \( O_2 \)
Explanation:
1. Masses of gas: A = 125 g, B = 60 g. Same heat and same \( \Delta T \): \( n_A\tfrac32R=n_B\tfrac52R \), so \( n_A/n_B=5/3 \).
2. \( \dfrac{125/M_A}{60/M_B}=\dfrac53 \), so \( M_B/M_A=0.8 \).
3. Monatomic A and diatomic B with ratio 0.8: Ar (40) and \( O_2 \) (32).
Teacher's Note:
1. Use the mole ratio from equal heat and equal temperature rise.
2. Check the pair against the list of molar masses.
(a) hydrogen, 1200 K
(b) nitrogen, 600 K
(c) hydrogen, 600 K
(d) oxygen, 1200 K
Show Answer & Explanation
Answer: (a) hydrogen, 1200 K
Explanation:
1. \( M=\dfrac{3RT}{v^2}=\dfrac{3(25/3)(300)}{1930^2}=2.0\times10^{-3} \) kg/mol, which is hydrogen.
2. Since \( v_{rms}\propto\sqrt T \), doubling the speed needs four times the temperature: 4 × 300 = 1200 K.
Teacher's Note:
1. Molar mass of 2 g/mol identifies H₂.
2. Speed doubles when the temperature becomes four times.
(a) \( V_1 \)
(b) \( (V_1+V_2)/2 \)
(c) \( \sqrt{V_1V_2} \)
(d) \( V_1/2 \)
Show Answer & Explanation
Answer: (a) \( V_1 \)
Explanation:
1. The average speed of a given gas depends only on its temperature and molar mass.
2. The temperature of C is T, so oxygen molecules in C move with the same average speed \( V_1 \) as in A.
Teacher's Note:
1. Molecules of one gas in a mixture are unaffected by the presence of the other gas at the same temperature.
2. Maxwell distributions of different gases are independent.
(a) \( P_0/2 \)
(b) \( 4P_0 \)
(c) \( 8P_0 \)
(d) \( 2P_0 \)
Show Answer & Explanation
Answer: (d) \( 2P_0 \)
Explanation:
1. \( P=\dfrac{1}{3}\dfrac NV m\overline{v^2} \).
2. The factor \( m\,v_{rms}^2 \) changes by \( \tfrac12\times4=2 \), so \( P=2P_0 \).
Teacher's Note:
1. Pressure depends on the product of molecular mass and the mean square speed.
2. The volume and number of molecules are unchanged.
(a) 0.48
(b) 0.448
(c) 0.032
(d) 0.512
Show Answer & Explanation
Answer: (b) 0.448
Explanation:
1. Initially \( n_1=\dfrac{PV}{RT}=\dfrac{3\times10^5\times0.004}{(25/3)(300)}=0.48 \) mol.
2. Finally \( n_2=\dfrac{10^5\times0.0008}{2500}=0.032 \) mol.
3. Leaked = 0.48 − 0.032 = 0.448 mol.
Teacher's Note:
1. After the puncture the pressure inside equals atmospheric pressure.
2. Use \( n=PV/RT \) before and after.
(a) 196 °C
(b) 112 °C
(c) 96 °C
(d) 292 °C
Show Answer & Explanation
Answer: (a) 196 °C
Explanation:
1. n = 0.040/4 = 0.01 mol.
2. \( \Delta U=n\tfrac32R\Delta T=0.01\times1.5\times\tfrac{25}{3}\Delta T=0.125\Delta T \), so \( \Delta T=96 \) K.
3. Final temperature = 100 + 96 = 196 °C.
Teacher's Note:
1. The container is closed (constant volume), so \( \Delta U=nC_v\Delta T \) with no work.
2. A temperature change in kelvin equals that in °C.
(a) v
(b) \( \sqrt2\,v \)
(c) 4v
(d) 2v
Show Answer & Explanation
Answer: (d) 2v
Explanation:
1. Average speed \( \propto\sqrt{T/M} \).
2. Temperature doubles and the molecular mass halves, so the factor is \( \sqrt{2/(1/2)}=2 \).
Teacher's Note:
1. Dissociation halves the mass of each particle.
2. The two effects multiply inside the square root.
(a) \( \dfrac{5}{6}P_0 \)
(b) \( P_0 \)
(c) \( \dfrac{6}{5}P_0 \)
(d) \( \dfrac{18}{13}P_0 \)
Show Answer & Explanation
Answer: (c) \( \dfrac{6}{5}P_0 \)
Explanation:
1. Moles before = 2 + 13 = 15; moles after = 10 + 8 = 18.
2. At constant V and T, \( P\propto n \), so \( P=\dfrac{18}{15}P_0=\dfrac65P_0 \).
Teacher's Note:
1. Count the gas moles on each side of the equation.
2. Water is treated as a gas here because the temperature is high.
(a) monatomic, monatomic
(b) diatomic, diatomic
(c) diatomic, monatomic
(d) monatomic, diatomic
Show Answer & Explanation
Answer: (d) monatomic, diatomic
Explanation:
1. Same volume, same heat and same temperature rise: \( \Delta P=\dfrac{nR\Delta T}{V} \) and \( Q=nC_v\Delta T \), so \( \Delta P\propto\dfrac{1}{C_v} \).
2. \( \dfrac{\Delta P_A}{\Delta P_B}=\dfrac{2.5}{1.5}=\dfrac{C_{vB}}{C_{vA}}=\dfrac{5}{3} \), which is \( \dfrac{5R/2}{3R/2} \).
3. So A is monatomic and B is diatomic.
Teacher's Note:
1. A smaller \( C_v \) gives a larger pressure rise for the same heat.
2. The ratio 5/3 matches the degrees of freedom of diatomic and monatomic gases.
(a) The kinetic energy of equal mass of gas
(b) The number of molecules in equal mass of gas
(c) The number of molecules in equal moles of gas
(d) The total kinetic energy of equal moles of gas
Show Answer & Explanation
Answer: (c) The number of molecules in equal moles of gas
Explanation:
1. Equal moles contain equal numbers of molecules (Avogadro's number per mole).
2. Total kinetic energy depends on the degrees of freedom, and equal masses contain different numbers of molecules, so the others are not universal.
Teacher's Note:
1. Avogadro's number is the same for all gases.
2. Only the translational energy per molecule is the same at a given temperature.
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