When analyzing mixtures of gases, such as the air we breathe, determining the exact proportion of each component gas is necessary. Since gases uniformly fill any container, scientists and engineers rely on measuring the force exerted by the gases on the container walls. This physical measurement offers a reliable way to quantify the amount of each substance present and determine the composition of the gas mixture.
Understanding Partial Pressure
The overall force a gas mixture exerts is the sum of the pressures generated by the individual gases within it. The pressure exerted by a single gas in a mixture is known as its partial pressure, represented by the symbol \(P_i\). This is the pressure the gas would exert if it were the only gas present in the container at the same temperature and volume. Because gas particles do not significantly interact in an ideal system, each gas acts independently of the others.
The total pressure, \(P_{total}\), of the entire mixture is the sum of all the partial pressures of the individual gases. For instance, in a container holding Gas A and Gas B, the total pressure is \(P_{total} = P_A + P_B\). Measuring partial pressure is a practical method for quantifying the amount of a gas in a lab or industrial setting. This measurement is relevant in fields like respiratory physiology, where the partial pressures of oxygen and carbon dioxide dictate gas exchange in the lungs.
Understanding Mole Fraction
To describe the composition of any mixture, including gases, the mole fraction (\(\chi_i\)) is used. The mole fraction is a ratio expressing the amount of one component relative to the total amount of all components present. It is calculated by dividing the number of moles of a single gas by the total number of moles of all gases in the mixture.
This ratio provides a standardized way to describe concentration without needing to specify a volume or temperature. The mole fraction is a dimensionless number that always falls between 0 and 1. A value of 0.25, for example, indicates that 25% of the total gas molecules belong to that specific component. This measure is useful because it remains constant even if the temperature or volume of the container changes.
The Relationship Between Pressure and Composition
The connection between the partial pressure of a gas and its concentration is a direct consequence of gas behavior in a mixture. For ideal gases, the pressure exerted by a gas is directly proportional to the number of gas molecules present. This means the fractional pressure contribution of a gas is identical to its fractional mole contribution. The partial pressure of a gas is the product of its mole fraction and the total pressure of the mixture.
Therefore, to find the mole fraction (\(\chi_i\)) of any gas, divide the partial pressure of that gas (\(P_i\)) by the total pressure of the mixture (\(P_{total}\)). The formula is \(\chi_i = P_i / P_{total}\). This mathematical link allows scientists to determine the composition of a gas mixture using pressure readings, which are easier to obtain than direct mole counts. This relationship holds true for ideal gases and serves as a close approximation for real gases under common conditions.
Calculating Mole Fraction: A Step-by-Step Example
Consider a simplified gas mixture containing only nitrogen and oxygen, with a measured total pressure of \(1.50\) atmospheres (atm). If the partial pressure exerted by the nitrogen gas (\(P_{N_2}\)) is \(1.17\) atm, the mole fraction of nitrogen can be calculated directly. The first step involves identifying the known values: \(P_{N_2} = 1.17\) atm and \(P_{total} = 1.50\) atm.
Next, apply the relationship \(\chi_{N_2} = P_{N_2} / P_{total}\). Substituting the measured values into the formula yields \(\chi_{N_2} = 1.17 \text{ atm} / 1.50 \text{ atm}\). The calculation results in a mole fraction value of \(0.78\). The units of pressure cancel out, confirming that mole fraction is a dimensionless quantity.
The resulting value of \(0.78\) indicates that nitrogen accounts for \(78\%\) of the total moles of gas in the mixture. Since the mixture contains only two gases, the mole fraction of oxygen (\(\chi_{O_2}\)) can be found by subtracting the nitrogen mole fraction from 1. Therefore, \(\chi_{O_2} = 1.00 – 0.78 = 0.22\), meaning oxygen makes up \(22\%\) of the mixture by moles.