Dalton's law of partial pressures Knowledge-Based Study

Study notes

1. In a mixture of non-reacting gases, each individual gas expands to fill the entire volume of the container and exerts its own characteristic pressure. This pressure is known as the partial pressure of that specific gas.

   According to the kinetic molecular theory, ideal gas particles are assumed to have negligible volume and experience no attractive or repulsive forces between one another. As a result, each gas in a mixture behaves independently of the other gases present, exerting the exact same pressure it would exert if it occupied the container alone at the same temperature.

   Formulated by John Dalton in 1801, Dalton's Law of Partial Pressures states that the total pressure exerted by a mixture of non-reacting gases is equal to the sum of the partial pressures of all the individual gases present in the mixture.

   Assuming a mixture containing gases 1, 2, 3, and so on at constant volume and temperature, Dalton's law is represented as:

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

   Applying the ideal gas law ($P = \frac{nRT}{V}$) to each gas component shows that:

   $P_{\text{total}} = (n_1 + n_2 + n_3 + \dots + n_n) \frac{RT}{V} = n_{\text{total}} \frac{RT}{V}$

   This relationship highlights that the total pressure depends solely on the total number of moles of gas particles, irrespective of the identity of the gases.

   The proportion of a particular gas in a mixture can be quantified using the mole fraction ($X$), which is the ratio of the moles of that component to the total moles of all components in the mixture.

   For a component gas $A$:

   $X_A = \frac{n_A}{n_{\text{total}}}$

   Because the sum of the moles of all components equals $n_{\text{total}}$, the sum of all mole fractions in a mixture always equals 1:

   $X_A + X_B + \dots + X_n = 1$

   By comparing the ideal gas equation for gas $A$ ($P_A = n_A \frac{RT}{V}$) to that of the total mixture ($P_{\text{total}} = n_{\text{total}} \frac{RT}{V}$), the partial pressure of component $A$ can be related directly to its mole fraction:

   $P_A = X_A \times P_{\text{total}}$

   A common practical application of Dalton's law is the laboratory collection of an insoluble or slightly soluble gas over water via displacement.

   As gas bubbles rise through the water and displace it, water molecules evaporate into the gas space until dynamic equilibrium is reached. Consequently, the collected gas is never pure; it is a mixture of the generated dry gas and water vapor.

   The total pressure of the wet gas mixture inside the collecting tube is:

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

   where $P_{\text{H}_2\text{O}}$ is the vapor pressure of water at the experimental temperature. To find the true pressure of the dry gas, the vapor pressure of water is subtracted from the total pressure:

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