Chapter 5.4: Gases – Mixtures, Dalton’s Law, and Stoichiometry of Gas Reactions

Mixtures of Gases

Composition of Air

Air is a mixture of several gases, with each component contributing to the overall properties of the atmosphere. The composition of dry air by volume is as follows:

Gas	Percent by Volume (%)
Nitrogen (N2)	78
Oxygen (O2)	21
Argon (Ar)	0.9
Carbon dioxide (CO2)	0.04

• Mixtures of gases behave similarly to pure gases, and their properties can be described using the same gas laws.

• Each gas in a mixture acts independently and exerts its own pressure, called partial pressure.

Dalton’s Law of Partial Pressures

Definition and Application

Dalton’s Law of Partial Pressures states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of individual gases.

• Partial pressure is the pressure that a gas in a mixture would exert if it occupied the entire volume alone at the same temperature.

• Mathematically, Dalton’s Law is expressed as:

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

• If water vapor is present, the total pressure is the sum of the dry gas pressure and the vapor pressure of water:

$P_{\text{humid air}} = P_{\text{dry air}} + P_{\text{water vapor}}$

• This law is essential for calculating the pressure of gases collected over water and in other laboratory settings.

Partial Pressure and Mole Fraction

Volume, Mass, and Mole Fractions

In mixtures, the contribution of each component can be described by its fraction of the total volume, mass, or number of moles.

• Volume fraction ($\varphi$): $\varphi_{\text{component}} = \frac{V_{\text{component}}}{V_{\text{total}}}$

• Mass fraction ($w$): $w_{\text{component}} = \frac{m_{\text{component}}}{m_{\text{total}}}$

• Mole fraction ($X$): $X_{\text{component}} = \frac{n_{\text{component}}}{n_{\text{total}}}$

Example: In a mixture of 1 mol O2 and 4 mol He (Heliox), the mole fraction of O2 is:

$X_{O_2} = \frac{1}{1+4} = 0.2$

• The partial pressure of a gas in a mixture is proportional to its mole fraction:

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

Dalton’s Law and Mole Fraction

Mathematical Derivation

• For a mixture of gases, the total pressure can be derived from the ideal gas law:

$P = \frac{nRT}{V}$

• For two gases, 1 and 2, in the same container:

$P_{\text{total}} = \frac{(n_1 + n_2)RT}{V}$

• The partial pressure of gas 1 is:

$P_1 = X_1 P_{\text{total}}$

• Where $X_1 = \frac{n_1}{n_1 + n_2}$ is the mole fraction of gas 1.

Applications of Dalton’s Law

Combining Gases and Calculating Final Pressure

• When two containers of gases are combined, the final pressure can be calculated using the total number of moles and the total volume:

$P_{\text{final}} = \frac{(n_{\text{H}_2} + n_{\text{He}})RT}{V_{\text{total}}}$

• Alternatively, sum the partial pressures if the gases are mixed in the same volume:

$P_{\text{final}} = P_{\text{H}_2,\text{mix}} + P_{\text{He},\text{mix}}$

Example Calculation

• Given: 1.25 mol H2 and 0.50 mol He in a 5.0 L container at 20°C.

• Calculate the total pressure:

$P_T = \frac{(1.25 + 0.50) \text{mol} \times 0.08314 \text{bar} \cdot \text{L} \cdot \text{mol}^{-1} \cdot \text{K}^{-1} \times 293.2 \text{K}}{5.0 \text{L}}$

Partial Pressure from Masses of Gases

Calculating Partial Pressure Using Mass and Molar Mass

• Given the mass of each gas, calculate the number of moles:

$n = \frac{m}{M}$

• Find the mole fraction for each gas:

$X_{\text{Ne}} = \frac{n_{\text{Ne}}}{n_{\text{Ar}} + n_{\text{Ne}} + n_{\text{CO}_2}}$

• Calculate the partial pressure:

$P_{\text{Ne}} = X_{\text{Ne}} P_T$

Chemical Reactions Over Water

Gas Collection and Water Vapor Correction

• When a gas is collected over water, the measured pressure includes both the gas and water vapor.

• The actual pressure of the collected gas is:

$P_{\text{gas}} = P_{\text{total}} - P_{\text{H}_2O}$

• Use the vapor pressure of water at the collection temperature to correct the measured pressure.

Example Table: Vapor Pressure of Water

Temperature (°C)	Pressure (Torr)
20	17.54
25	23.78
30	31.82
35	42.18
40	55.36

Stoichiometry of Chemical Reactions with Gases

Using Gas Laws in Chemical Calculations

• Gas laws can be combined with stoichiometry to determine the amounts of reactants or products in reactions involving gases.

• At standard temperature and pressure (STP: 0°C, 1 atm), 1 mol of an ideal gas occupies 22.4 L.

• For reactions producing or consuming gases, use the ideal gas law to relate volume, pressure, temperature, and moles:

$PV = nRT$

• Example: Decomposition of sodium azide (NaN3) in airbags:

$2\text{NaN}_3(s) \rightarrow 2\text{Na}(s) + 3\text{N}_2(g)$

• Calculate the mass of NaN3 needed to produce a certain volume of N2 at STP.

Key Concepts

• Understanding partial pressures and mole fractions in gas mixtures is essential for predicting the behavior of gases in chemical and physical processes.

• Dalton’s Law is fundamental for calculating the pressure of gases collected over water and in laboratory experiments.

• Stoichiometry involving gases requires the use of the ideal gas law to relate moles, volume, pressure, and temperature.