Gas Laws: Total Pressure, Partial Pressures, and Gas Collection Over Water

Gas Laws and Mixtures of Gases

Total Pressure and Partial Pressures

In a mixture of gases, the total pressure is the sum of the partial pressures of each individual gas. This principle is described by Dalton's Law of Partial Pressures, which is fundamental in understanding the behavior of gas mixtures.

• Partial Pressure (Pi): The pressure exerted by a single component in a mixture of gases.

• Dalton's Law: The total pressure of a mixture of gases is equal to the sum of the partial pressures of the individual gases.

Equation:

$P_{total} = P_1 + P_2 + P_3 + ext{...}$

• Mole Fraction (Xi): The ratio of the number of moles of a component to the total number of moles in the mixture.

Equation:

$X_i = \frac{n_i}{n_{total}}$

• Partial Pressure from Mole Fraction: The partial pressure of a gas can be calculated using its mole fraction and the total pressure.

Equation:

$P_i = X_i \times P_{total}$

Example Calculation

• Given a mixture of helium, neon, and argon with known masses and total pressure, you can calculate:

  • The number of moles of each gas using their molar masses.

  • The mole fraction of each gas.

  • The partial pressure of each gas using Dalton's Law.

• Convert mass to moles: $n = \frac{m}{M}$

• Calculate mole fraction: $X_i = \frac{n_i}{n_{total}}$

• Calculate partial pressure: $P_i = X_i \times P_{total}$

Collecting a Gas Over Water

When a gas is collected over water, the total pressure inside the collection vessel is the sum of the pressure of the gas and the vapor pressure of water. The vapor pressure of water depends on temperature and must be subtracted from the total pressure to obtain the pressure of the dry gas.

• Vapor Pressure of Water (PH2O): The pressure exerted by water vapor at a given temperature.

• Collected Gas Pressure: $P_{gas} = P_{total} - P_{H2O}$

Equation:

$P_{total} = P_{gas} + P_{H_2O}$

Example Table: Vapor Pressure of Water at Various Temperatures

Temperature (°C)	Pressure (mmHg)
5.0	6.54
10.0	9.21
15.0	12.79
20.0	17.54
25.0	23.76
30.0	31.82
35.0	42.18
40.0	55.32
50.0	92.51
60.0	149.4
70.0	233.7
80.0	355.1
90.0	525.8
100.0	760.0

Example: Calculating the Mass of Oxygen Collected Over Water

• Given: Volume of gas collected, total pressure, temperature.

• Find: Mass of oxygen gas collected.

• Steps:

  1. Subtract vapor pressure of water from total pressure to get pressure of oxygen.

  2. Use the ideal gas law to find moles of oxygen: $n = \frac{PV}{RT}$

  3. Convert moles to grams using molar mass.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas.

Equation:

$PV = nRT$

• P: Pressure (atm)

• V: Volume (L)

• n: Number of moles

• R: Ideal gas constant ($0.0821 \text{ L atm mol}^{-1} \text{K}^{-1}$)

• T: Temperature (K)

Summary Table: Key Relationships in Gas Mixtures

Relationship	Equation	Description
Dalton's Law	$P_{total} = \sum P_i$	Total pressure is sum of partial pressures
Mole Fraction	$X_i = \frac{n_i}{n_{total}}$	Fraction of moles of a component
Partial Pressure	$P_i = X_i \times P_{total}$	Pressure of a component in mixture
Ideal Gas Law	$PV = nRT$	Relates P, V, n, T for a gas
Gas over Water	$P_{gas} = P_{total} - P_{H_2O}$	Pressure of dry gas collected over water

Applications and Examples

• Calculating Partial Pressures: Used in chemical reactions involving gas mixtures, such as in stoichiometry and equilibrium calculations.

• Collecting Gases Over Water: Common laboratory technique for measuring the volume and pressure of gases produced in reactions.

• Determining Gas Composition: Analysis of air samples, industrial gas mixtures, and respiratory gases.

Example: If a mixture contains 0.16 mol He, 0.34 mol Ne, and 1.00 mol Ar at a total pressure of 662 mmHg, the partial pressure of each gas can be calculated using their mole fractions and Dalton's Law.

Additional info: These notes expand on the examples and tables shown in the images, providing context and formulas for exam preparation in General Chemistry.