Question 1
Estimate the fraction of the volume occupied by oxygen molecules compared with the gas volume at STP. Take the molecular diameter as 3 Å.
Steps only
[Nₐ(4πr³/3)]/(2.24×10⁻²)=3.8×10⁻⁴
Answer
3.8×10⁻⁴
CLASS 11 · PHYSICS · CHAPTER 12
Kinetic theory
Simple solutions to all 10 end-of-chapter questions on gas laws, molecular motion, energy and mean free path.
10 worked answersComplete chapter answers
10 worked answers · short steps first, explanations when you need them
Chapter 12 Exercises
Textbook pages 255–256
Question 2
Show that one mole of an ideal gas occupies 22.4 L at STP (1 atm and 0°C).
Steps only
V=nRT/P=(8.31×273)/(1.01×10⁵)=0.0224 m³=22.4 L
Answer
22.4 L
Question 3
For 1.00×10⁻³ kg of oxygen, interpret the PV/T versus P graph: (a) dotted line, (b) compare T₁ and T₂, (c) find the common y-intercept, (d) find the hydrogen mass with the same low-pressure intercept.
Steps only
At low pressure PV/T=nR. Oxygen has n=0.03125 mol, so the intercept is 0.260 J K⁻¹; the same n needs 0.063 g of H₂.
Answer
(a) Ideal-gas line; (b) T₁>T₂; (c) 0.260 J K⁻¹; (d) 0.063 g H₂
Question 4
A 30 L oxygen cylinder cools from 27°C to 17°C as its gauge pressure falls from 15 atm to 11 atm. Estimate the mass of oxygen withdrawn. Use R=8.31 J mol⁻¹ K⁻¹ and molar mass 32 g/mol.
Steps only
Δn=(0.030×1.013×10⁵/8.31)(16/300−12/290)≈4.4 mol; m≈4.4×32=140 g
Answer
About 0.14 kg
Question 5
A 1.0 cm³ air bubble rises from a lake 40 m deep. Water is 12°C at the bottom and 35°C at the surface. Find the bubble’s surface volume.
Steps only
V₂=1.0×(4.93×10⁵/1.013×10⁵)×(308/285)≈5.3 cm³
Answer
About 5.3 cm³
Question 6
Estimate the number of air molecules in a 25.0 m³ room at 27°C and 1 atm.
Steps only
N=PV/(kBT)=(1.013×10⁵×25.0)/(1.38×10⁻²³×300)≈6.1×10²⁶
Answer
6.1×10²⁶ molecules
Question 7
Estimate the average thermal energy of a helium atom at (i) 27°C, (ii) 6000 K and (iii) 10 million K.
Steps only
E=(3/2)kBT gives 6.2×10⁻²¹ J, 1.24×10⁻¹⁹ J and 2.07×10⁻¹⁶ J, respectively.
Answer
- (i) 6.2×10⁻²¹ J;
- (ii) 1.24×10⁻¹⁹ J;
- (iii) 2.07×10⁻¹⁶ J
Question 8
Equal-capacity vessels contain neon, chlorine and uranium hexafluoride at the same pressure and temperature. Do they have equal molecule counts? Compare their rms speeds.
Steps only
PV=NkBT gives equal N. Since vᵣₘₛ∝1/√M, neon (the lightest) is fastest.
Answer
Equal counts; neon has the greatest rms speed.
Question 9
At what temperature will argon have the same rms speed as helium at −20°C? Atomic masses: Ar=39.9 u, He=4.0 u.
Steps only
TAr=253×(39.9/4.0)≈2.52×10³ K≈2.25×10³°C
Answer
2.52×10³ K (about 2.25×10³°C)
Question 10
Estimate the mean free path and collision frequency of nitrogen at 2.0 atm and 17°C. The molecular radius is 1.0 Å. Compare collision duration with the free time between collisions. Molar mass of N₂=28.0 g/mol.
Steps only
n=P/(kBT)≈5.1×10²⁵ m⁻³; λ=1/(√2nπd²)≈1.7×10⁻⁷ m; f≈vᵣₘₛ/λ≈3.0×10⁹ s⁻¹
Answer
λ≈1.7×10⁻⁷ m; f≈3.0×10⁹ s⁻¹