Theory
Formulated by John Dalton in 1801, Dalton's Law of Partial Pressures states that in a mixture of non-reacting gases, the total pressure exerted by the mixture is equal to the sum of the partial pressures that each individual gas would exert if it uniquely occupied the entire volume of the container at an identical thermodynamic temperature.
This principle assumes ideal gas behavior, implying that the constituent gas molecules experience negligible intermolecular forces and their volumes are infinitesimally small relative to the total volume of the system.
Mathematical Expression
For a homogeneous mixture containing \(n\) distinct, non-interacting gases, the law is defined as:
$$P_{\text{total}} = P_1 + P_2 + P_3 + \dots + P_n$$Where:
- \(P_{\text{total}}\) represents the aggregate pressure of the gaseous mixture.
- \(P_1, P_2, P_n\) represent the independent partial pressures of the respective component gases.
Derivation via the Ideal Gas Law
The validity of Dalton’s Law can be verified by applying the ideal gas equation (\(PV = nRT\)) to a mixture confined within a fixed volume (\(V\)) at a constant absolute temperature (\(T\)).
If the mixture contains distinct molar quantities of gases (\(n_1, n_2, \dots, n_n\)), the total number of moles (\(n_{\text{total}}\)) is given by:
$$n_{\text{total}} = \sum_{i=1}^{n} n_i = n_1 + n_2 + \dots + n_n$$Substituting the aggregate molar quantity into the ideal gas equation yields:
$$P_{\text{total}} = \frac{n_{\text{total}}RT}{V} = \frac{(n_1 + n_2 + \dots + n_n)RT}{V}$$Expanding the algebraic terms gives:
$$P_{\text{total}} = \frac{n_1 RT}{V} + \frac{n_2 RT}{V} + \dots + \frac{n_n RT}{V}$$Because the partial pressure of an individual component \(i\) is defined as \(P_i = \frac{n_i RT}{V}\), substitution recovers the fundamental expression:
$$P_{\text{total}} = P_1 + P_2 + \dots + P_n$$Determining Partial Pressures Using Mole Fraction
In practical laboratory and industrial scenarios, partial pressures are frequently calculated using the mole fraction (\(\chi\)), which represents the ratio of the number of moles of a specific component to the total moles of gas in the system.
The mole fraction of a generic component \(i\) is defined as:
$$\chi_i = \frac{n_i}{n_{\text{total}}}$$By taking the ratio of the individual partial pressure equation to the total pressure equation, the volumetric and thermal variables cancel out:
$$\frac{P_i}{P_{\text{total}}} = \frac{\frac{n_i RT}{V}}{\frac{n_{\text{total}}RT}{V}} = \frac{n_i}{n_{\text{total}}} = \chi_i$$Rearranging this yields the primary analytical formula for determining partial pressures:
$$P_i = \chi_i \cdot P_{\text{total}}$$Practical Laboratory Application: Vapor Pressure Compensation
A classic application of Dalton’s Law occurs when a gas is generated via a chemical reaction and collected over water via downward displacement. The gas sample becomes saturated with water vapor, turning it into a binary mixture.
To isolate the true partial pressure of the dry gas (\(P_{\text{gas}}\)), the vapor pressure of water (\(P_{\text{H}_2\text{O}}\))—which is strictly a function of the system's temperature—must be subtracted from the total barometric pressure (\(P_{\text{total}}\)):
$$P_{\text{total}} = P_{\text{gas}} + P_{\text{H}_2\text{O}}$$ $$P_{\text{gas}} = P_{\text{total}} - P_{\text{H}_2\text{O}}$$This fundamental adjustment is critical for accurate calculations in stoichiometric and volumetric analysis.
Related Topic Dalton's Atomic Theory