Partial pressure is a fundamental concept in chemistry and atmospheric science, defining the individual pressure exerted by a specific gas within a mixture of gases. Air, for example, is composed primarily of nitrogen, oxygen, and argon, and the total atmospheric pressure is the sum of the pressures contributed by each component. Understanding this concept allows scientists to analyze gas behavior in diverse environments, from scuba tanks and medical ventilators to industrial reactions.
The Foundation: Dalton’s Law of Partial Pressures
The framework for calculating partial pressure rests upon Dalton’s Law of Partial Pressures. This law states that the total pressure exerted by a mixture of non-reacting gases is equal to the sum of the partial pressures of the individual gases. Every gas in the container acts independently, contributing its own pressure to the overall system pressure.
This independent behavior occurs because gas particles are separated by vast distances relative to their size. Consequently, the forces of attraction or repulsion between different gas molecules are considered negligible, meaning the introduction of one gas does not interfere with the behavior of another. The pressure each gas exerts is solely determined by the frequency and force of its own particles’ collisions with the container walls.
The fundamental mathematical expression of this concept is straightforward, showing that the total pressure (\(P_{total}\)) is the sum of the pressures of gas 1 (\(P_1\)), gas 2 (\(P_2\)), and so on. For instance, in a medical oxygen tank containing a mixture of oxygen and helium, the gauge reading for total pressure is simply the sum of the individual pressures contributed by the oxygen molecules and the helium atoms. This foundational principle allows researchers to isolate the influence of one specific gas within a complex mixture, provided the pressures of all other components are known.
Calculating Partial Pressure Using Mole Fraction
While Dalton’s Law explains the additivity of pressures, the most common method for calculating an individual partial pressure relies on knowing the composition of the gas mixture. This calculation uses the concept of the mole fraction, which precisely quantifies the relative amount of a specific gas within the total system. The mole fraction (\(X_A\)) of a gas A is defined as the number of moles of gas A (\(n_A\)) divided by the total number of moles of all gases in the mixture (\(n_{total}\)).
The relationship between partial pressure and mole fraction is direct: the partial pressure of a gas A (\(P_A\)) is equal to its mole fraction (\(X_A\)) multiplied by the total pressure of the gas mixture (\(P_{total}\)). This connection arises from the ideal gas law, which demonstrates that at a constant temperature and volume, the pressure exerted by a gas is directly proportional to the number of moles present. If a gas constitutes 25% of the total moles, it will contribute exactly 25% of the total system pressure.
To illustrate this calculation, consider a sealed tank containing 2.0 moles of nitrogen gas (\(N_2\)) and 6.0 moles of oxygen gas (\(O_2\)), where the total measured pressure is 4.0 atmospheres (atm). The first step is to determine the total number of moles (8.0 total moles). Next, the mole fraction of nitrogen (\(X_{N_2}\)) is calculated by dividing the moles of nitrogen (2.0) by the total moles (8.0), resulting in a mole fraction of 0.25.
The final step involves multiplying this mole fraction by the total pressure of 4.0 atm to find the partial pressure of nitrogen. The result is \(0.25 \times 4.0 \text{ atm}\), which equals 1.0 atm. This method is effective because the mole fraction remains constant regardless of typical changes in temperature or volume, provided the system is closed. This calculation is widely applied in industrial processes and medical settings.
A Practical Application: Gases Collected Over Water
A common scenario in introductory chemistry involves collecting a gas product by displacing water from an inverted container. In this experimental setup, the gas collected is not pure; it is a mixture of the desired dry gas and water vapor. Since the gas is collected over a liquid, some of the water evaporates into the container’s headspace, contributing its own pressure to the total measured pressure.
When a manometer measures the total pressure (\(P_{total}\)) of the gas mixture in the collection vessel, it is measuring the sum of the partial pressure of the dry gas (\(P_{gas}\)) and the partial pressure of the water vapor (\(P_{water vapor}\)). To accurately determine the pressure of the gas of interest, one must use Dalton’s law to subtract the contribution of the water vapor from the total pressure. The adjusted formula becomes: \(P_{gas} = P_{total} – P_{water vapor}\).
The pressure exerted by the water vapor is known as the vapor pressure of water, and its value is entirely dependent on the temperature of the water. As the temperature increases, more water molecules possess enough kinetic energy to escape into the gaseous phase, causing the vapor pressure to rise significantly. Therefore, accurately measuring the temperature of the water is necessary to look up the corresponding vapor pressure value from established reference tables, allowing for the precise calculation of the dry gas’s partial pressure.