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Kinetic Theory of Gases · Part 2 of 9

Gas Laws

Boyle, Charles and Gay-Lussac each found one simple rule for a fixed amount of gas. Put together, they give the ideal gas equation PV = nRT, the tool for almost every problem in this chapter.

Builds on: Part 1 · Postulates of Kinetic Theory.

Video coming soon

Boyle's law

$P V = constant$ at constant T

Push a piston in slowly at constant temperature: halve the volume and the pressure doubles. For a fixed amount of gas, $P V = constant$ . It fails near liquefaction, where the gas stops behaving ideally.

A cylinder of gas with a piston pushed down. — Constant temperature: smaller volume, more wall hits, higher pressure.

Three P–V hyperbolas for increasing temperatures. — T₃ > T₂ > T₁.

On a graph

Isotherms are rectangular hyperbolas

At each temperature, P against V is a rectangular hyperbola $P V = constant$ . Hotter isotherms lie further from the axes.

Charles and Gay-Lussac

Every line meets −273.15 °C

At constant pressure (a freely moving weighted piston, so $P_{gas} = P_{atm} + \frac{M g}{S}$ ), volume rises linearly with temperature in °C. Extend the lines for any gas and they all reach V = 0 at −273.15 °C. That point is absolute zero, the start of the Kelvin scale.

Straight V–t lines extrapolated to −273.15 °C. — Different gases and amounts: the same intercept.

V–T lines through the origin. — With T in kelvin, V ∝ T.

Kelvin scale

V ∝ T in kelvin

Measured in kelvin, the lines pass through the origin: $\frac{V}{T} = constant$ at constant P, and $\frac{P}{T} = constant$ at constant V. Always use kelvin in gas-law formulas.

Dalton's law

Partial pressures add

In a mixture, each gas exerts the pressure it would alone in the whole volume: $P_{total} = P_{1} + P_{2} + \dots$ . Mean molar mass: $M_{mix} = \frac{\sum n _{i} M _{i}}{\sum n _{i}}$ . Semipermeable walls let only some gases through, so each part's pressure is the sum of the gases that reach it.

A vessel in three parts with semipermeable partitions. — A vessel split by semipermeable partitions.

Summary

Key formulas

P_{1} V_{1} = P_{2} V_{2}

Boyle's law

\frac{V _{1}}{V _{2}} = \frac{T _{1}}{T _{2}}

Charles and Gay-Lussac's law

\frac{V}{T} = constant

Charles and Gay-Lussac's law

P V = n R T

Ideal gas equation

P V = N k T

Ideal gas equation

n = \frac{N}{N _{A}} = \frac{m}{M}

Ideal gas equation

P = \frac{ρR T}{M}

Ideal gas equation

N_{A} = 6.023 \times 1 0^{23} mo l^{- 1}

Avogadro's number and hypothesis

k = \frac{R}{N _{A}} = 1.38 \times 1 0^{- 23} J/K

Avogadro's number and hypothesis

P_{total} = P_{1} + P_{2} + \dots + P_{n}

Dalton's law of partial pressures

M_{mix} = \frac{\sum n _{i} M _{i}}{\sum n _{i}}

Dalton's law of partial pressures

Worked examples

From the video

1. Volume of one mole of gas at STP (1 atm, 273 K).

$V = \frac{R T}{P} = \frac{8.314 \times 273}{101325}$ ≈ 22.4 L

2. A 30 L vessel holds gas at 63 °C. After some gas escapes, the pressure drops by 0.415 bar at constant T. Density at STP (1 atm) is 1.3 g/L. Find the mass released.

$Δ n = \frac{Δ P V}{R T} = \frac{0.415 \times 1 0 ^{5} \times 0.030}{8.314 \times 336}$ ≈ 0.446 mol. $M = 1.3 g/L \times 22.4 L/mol$ ≈ 29.1 g/mol, so Δm ≈ 12.98 g. (13.14 g comes from taking STP as 1 bar.)

3. Two bulbs of 3 L and 1 L joined by a capillary hold air at 76 cm Hg and 30 °C. The 3 L bulb is put in steam at 100 °C; the other stays at 30 °C. New pressure?

Moles are conserved: $P (\frac{3}{373} + \frac{1}{303}) = \frac{76 \times 4}{303}$ → P ≈ 88.45 cm Hg

4. A free piston divides a closed vertical cylinder into two parts, each with 1 mol of air. At 300 K the upper volume is 4 times the lower. At what temperature will the ratio be 2?

The piston's weight gives a fixed pressure difference: $P_{lower} - P_{upper} = R T (\frac{1}{V _{L}} - \frac{1}{V _{U}})$ stays constant. Solving with $V_{U} / V_{L} = 4$ at 300 K and 2 at T gives T = 750 K.

5. An 8.3 L vessel at 300 K holds 0.1 mol N₂, 0.2 mol O₂ and 0.3 mol CO₂. Pressure and mean molar mass?

$P = \frac{n R T}{V} = \frac{0.6 \times 8.314 \times 300}{8.3 \times 1 0 ^{- 3}}$ ≈ 1.80 × 10⁵ Pa; $M_{mix} = \frac{2.8 + 6.4 + 13.2}{0.6}$ ≈ 37.33 g/mol

6. A 30 L vessel at 300 K is split into three equal parts holding 30 g H₂, 160 g O₂ and 70 g N₂. The left partition passes only H₂; the right passes H₂ and N₂. Find the pressure in each part.

H₂ spreads through all 30 L; N₂ through the middle and right (20 L); O₂ stays in the middle (10 L). P₁ = 12.47 × 10⁵ Pa, P₂ = 28.06 × 10⁵ Pa, P₃ = 15.59 × 10⁵ Pa

JEE-style question

Your turn

An ideal gas at 27 °C is heated at constant pressure until its volume doubles. Find the final temperature.

1)327 °C

2)54 °C

3)600 °C

4)327 K

Show the answer and the traps

V/T is constant, in kelvin: 300 K doubles to 600 K, which is 327 °C. Option 1.

Option 2, 54 °C, doubles the Celsius value. Options 3 and 4 mix up kelvin and Celsius.

Watch out

Common mistakes

Using °C in PV = nRT.Temperatures must be in kelvin: T = t + 273.15.

Using M in g/mol in P = ρRT/M.In SI units M is in kg/mol (O₂: 0.032).

Forgetting the piston's weight.A weighted free piston gives

P_{gas} = P_{atm} + \frac{M g}{S}

, not P_atm.

Practice

Try these

1. A gas at 1 atm and 27 °C occupies 2 L. It is heated to 127 °C and compressed to 1 L. New pressure?

$P_{2} = \frac{P _{1} V _{1} T _{2}}{V _{2} T _{1}} = \frac{1 \times 2 \times 400}{1 \times 300}$ ≈ 2.67 atm.

2. Volume of 1 mol of an ideal gas at 27 °C and 1 atm?

$V = \frac{R T}{P} = \frac{8.314 \times 300}{101325}$ ≈ 2.46 × 10⁻² m³ = 24.6 L.

3. Density of O₂ at 2 atm and 300 K?

$ρ = \frac{P M}{R T} = \frac{2 \times 101325 \times 0.032}{8.314 \times 300}$ ≈ 2.60 kg/m³.

4. 2 g of H₂ and 28 g of N₂ fill a 10 L vessel at 300 K. Total pressure, partial pressures and mean molar mass?

1 mol each: $P = \frac{2 \times 8.314 \times 300}{0.01}$ ≈ 4.99 × 10⁵ Pa; each partial pressure ≈ 2.49 × 10⁵ Pa; $M_{mix} = \frac{30}{2} = 15$ g/mol.

Gas Laws

PV=constant at constant T

Isotherms are rectangular hyperbolas

Every line meets −273.15 °C

V ∝ T in kelvin

Partial pressures add

Key formulas

From the video

Your turn

Common mistakes

Try these

$P V = constant$ at constant T