Core Concept

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 each individual gas.

Dalton’s Law Formula

The total pressure of a gas mixture can be represented as:

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

Where:

• $P_{\text{total}}$​= Total pressure of the gas mixture.

• $P_1, P_2, P_3, \ldots$ = Partial pressures of each gas in the mixture.

Each partial pressure is the pressure that each gas would exert if it occupied the entire volume of the container alone.

Key Concepts

1. Partial Pressure:

   • The pressure exerted by a single gas in a mixture.

   • Each gas in a mixture behaves independently, and its pressure is proportional to its mole fraction in the mixture.

2. Mole Fraction (X):

   • The mole fraction of a gas is the ratio of the moles of that gas to the total moles of all gases in the mixture.

   • Mole fraction (X) is calculated as: $X_i = \frac{n_i}{n_{\text{total}}}$

   • Where $X_i$​ is the mole fraction of gas i, $n_i$​ is the moles of gas i, and $n_{\text{total}}$​ is the total moles of all gases.

3. Relationship Between Partial Pressure and Mole Fraction:

   • The partial pressure of each gas in a mixture is directly proportional to its mole fraction and the total pressure: $P_i = X_i \cdot P_{\text{total}}$

   • This means that if you know the mole fraction of a gas and the total pressure, you can calculate its partial pressure.

Applications of Dalton’s Law

1. Collecting Gas Over Water:

   • When collecting a gas over water, the total pressure includes both the pressure of the gas and the vapor pressure of water.

   • The total pressure is: $P_{\text{total}} = P_{\text{gas}} + P_{\text{water vapor}}$

   • To find the pressure of the gas alone, you subtract the vapor pressure of water (often provided in tables at different temperatures) from the total pressure.

2. Gas Mixtures in Real-Life Applications:

   • Dalton’s Law is useful in applications where gas mixtures are present, such as atmospheric pressure (a mixture of nitrogen, oxygen, and other gases) and respiratory gases in medical applications.

Example Problem: Calculating Total Pressure

Problem: A container holds a mixture of three gases with partial pressures of $P_1$ = 2.0, $P_2$ = 1.5atm, and $P_3$ = 0.5atm. What is the total pressure in the container?

Solution:

$P_{\text{total}} = P_1 + P_2 + P_3 = 2.0 \, \text{atm} + 1.5 \, \text{atm} + 0.5 \, \text{atm} = 4.0 \, \text{atm}$

Answer: The total pressure is 4.0 atm.

Example Problem: Finding Partial Pressure Using Mole Fraction

Problem: A mixture of gases has a total pressure of 3.0 atm. If oxygen makes up 40% of the mixture by moles, what is the partial pressure of oxygen?

Solution:

1. Calculate the Mole Fraction of Oxygen:

   $X_{\text{O}_2} = 0.40$

2. Use Dalton’s Law to Find the Partial Pressure:

   $P_{\text{O}_2} = X_{\text{O}_2} \cdot P_{\text{total}} = 0.40 \times 3.0 \, \text{atm} = 1.2 \, \text{atm}$

Answer: The partial pressure of oxygen is 1.2 atm.

Example 3: Gas Collected Over Water

Problem: A gas is collected over water at 25∘C25^\circ\text{C}25∘C and the total pressure is 750 mmHg750 \, \text{mmHg}750mmHg. The vapor pressure of water at 25∘C25^\circ\text{C}25∘C is 24 mmHg24 \, \text{mmHg}24mmHg. What is the pressure of the gas?

Solution:

1. Use Dalton’s Law to Subtract Water Vapor Pressure: Pgas=Ptotal−Pwater vaporP_{\text{gas}} = P_{\text{total}} - P_{\text{water vapor}}Pgas​=Ptotal​−Pwater vapor​ Pgas=750 mmHg−24 mmHg=726 mmHgP_{\text{gas}} = 750 \, \text{mmHg} - 24 \, \text{mmHg} = 726 \, \text{mmHg}Pgas​=750mmHg−24mmHg=726mmHg

Answer: The pressure of the gas is 726 mmHg726 \, \text{mmHg}726mmHg.