Gas Laws

Overview

Gas laws describe the relationships between pressure, volume, temperature, and amount of gas. These relationships help predict gas behavior under changing conditions.

Standard Conditions

Condition	STP	SATP
Temperature	273.15 K (0°C)	298.15 K (25°C)
Pressure	1 atm (101.325 kPa)	1 bar (100 kPa)
Molar Volume	22.4 L/mol	24.8 L/mol

Pressure Units

Unit	Abbreviation	Conversion
Atmosphere	atm	1 atm = 101.325 kPa
Pascal	Pa	101,325 Pa = 1 atm
Kilopascal	kPa	101.325 kPa = 1 atm
Millimeters of mercury	mmHg	760 mmHg = 1 atm
Torr	torr	760 torr = 1 atm
Bar	bar	1.01325 bar = 1 atm

Temperature Conversion

$$K = °C + 273.15$$

Always use Kelvin in gas law calculations!

The Gas Laws

Boyle's Law (Constant T and n)

Pressure and volume are inversely proportional.

$$P_1V_1 = P_2V_2$$

Charles's Law (Constant P and n)

Volume and temperature are directly proportional.

$$\frac{V_1}{T_1} = \frac{V_2}{T_2}$$

Gay-Lussac's Law (Constant V and n)

Pressure and temperature are directly proportional.

$$\frac{P_1}{T_1} = \frac{P_2}{T_2}$$

Avogadro's Law (Constant T and P)

Volume and amount are directly proportional.

$$\frac{V_1}{n_1} = \frac{V_2}{n_2}$$

Combined Gas Law

$$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$$

Summary Table

Law	Constant	Relationship	Equation
Boyle's	T, n	$P \propto 1/V$	$P_1V_1 = P_2V_2$
Charles's	P, n	$V \propto T$	$V_1/T_1 = V_2/T_2$
Gay-Lussac's	V, n	$P \propto T$	$P_1/T_1 = P_2/T_2$
Avogadro's	T, P	$V \propto n$	$V_1/n_1 = V_2/n_2$

Examples

Example 1: Boyle's Law

A gas at 2.0 atm occupies 5.0 L. What is the volume at 4.0 atm?

$$P_1V_1 = P_2V_2$$ $$(2.0)(5.0) = (4.0)(V_2)$$ $$V_2 = \frac{10.0}{4.0} = 2.5 \text{ L}$$

Example 2: Charles's Law

A gas occupies 3.0 L at 300 K. What is the volume at 450 K?

$$\frac{V_1}{T_1} = \frac{V_2}{T_2}$$ $$\frac{3.0}{300} = \frac{V_2}{450}$$ $$V_2 = \frac{3.0 \times 450}{300} = 4.5 \text{ L}$$

Example 3: Gay-Lussac's Law

A gas at 1.5 atm and 27°C is heated to 127°C. What is the new pressure?

Convert to Kelvin: $T_1 = 300$ K, $T_2 = 400$ K

$$\frac{P_1}{T_1} = \frac{P_2}{T_2}$$ $$\frac{1.5}{300} = \frac{P_2}{400}$$ $$P_2 = \frac{1.5 \times 400}{300} = 2.0 \text{ atm}$$

Example 4: Combined Gas Law

A gas at 1.0 atm, 5.0 L, and 25°C is changed to 2.0 atm and 50°C. Find the new volume.

$T_1 = 298$ K, $T_2 = 323$ K

$$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$$ $$\frac{(1.0)(5.0)}{298} = \frac{(2.0)(V_2)}{323}$$ $$V_2 = \frac{1.0 \times 5.0 \times 323}{298 \times 2.0} = 2.7 \text{ L}$$

Dalton's Law of Partial Pressures

The total pressure of a gas mixture equals the sum of partial pressures.

$$P_{\text{total}} = P_1 + P_2 + P_3 + \cdots$$

Partial Pressure

$$P_i = X_i \times P_{\text{total}}$$

Where $X_i$ = mole fraction = $n_i/n_{\text{total}}$

Example

A mixture contains 2.0 mol N₂ and 1.0 mol O₂ at 3.0 atm total.

$$X_{\text{N}_2} = \frac{2.0}{3.0} = 0.67$$ $$X_{\text{O}_2} = \frac{1.0}{3.0} = 0.33$$ $$P_{\text{N}_2} = 0.67 \times 3.0 = 2.0 \text{ atm}$$ $$P_{\text{O}_2} = 0.33 \times 3.0 = 1.0 \text{ atm}$$

Collecting Gas Over Water

When collecting gas over water, account for water vapor pressure:

$$P_{\text{gas}} = P_{\text{total}} - P_{\text{H}_2\text{O}}$$

The vapor pressure of water depends on temperature (from tables).

Temp (°C)	$P_{\text{H}_2\text{O}}$ (mmHg)
20	17.5
25	23.8
30	31.8

Graham's Law of Effusion

Rate of effusion is inversely proportional to the square root of molar mass.

$$\frac{\text{Rate}_1}{\text{Rate}_2} = \sqrt{\frac{M_2}{M_1}}$$

Example

Compare effusion rates of H₂ and O₂.

$$\frac{\text{Rate}_{\text{H}_2}}{\text{Rate}_{\text{O}_2}} = \sqrt{\frac{32}{2}} = \sqrt{16} = 4$$

H₂ effuses 4 times faster than O₂.