- The mole fraction is a dimensionless ratio defined as the partial pressure of a gas divided by the total pressure, or the moles of a gas divided by total moles.
- No numerical data for pressures or molar quantities is provided in the problem, making a numerical solution impossible.
- The unit "atm" provided in the question is inappropriate for mole fraction, as it is a dimensionless value.
- To solve this, specific experimental data such as individual partial pressures or mole counts are required to perform the division \(\frac{P_i}{P_T}\) or \(\frac{n_i}{n_T}\).
Explanation
- Analysis of Provided Data
The problem presents two fundamental relationships in gas chemistry: Dalton's Law of Partial Pressures, which states that the total pressure of a gas mixture, \(P_T\), is the sum of the partial pressures of its individual components, \(P_T = P_1 + P_2 + \ldots + P_n\), and the definition of the mole fraction, \(X_a = \frac{n_a}{n_T} = \frac{P_a}{P_T}\), which relates the proportion of moles of a component to its partial pressure relative to the total pressure. However, the problem provides no numerical values for partial pressures, total pressure, or the number of moles for any gas component. Therefore, it is impossible to calculate a numerical result for the mole fractions of \(O_2\) or Ar with the information currently provided.
- Methodological Requirements and Clarifications
It is important to note that the mole fraction is a dimensionless quantity, as it represents the ratio of moles of a specific component to the total number of moles in a mixture. Consequently, the unit "atm" specified in the question is conceptually incorrect for this measure. To determine the mole fraction of a gas like \(O_2\), one must know either the partial pressure of that gas and the total pressure, or the individual number of moles of that gas and the total number of moles of all gases present in the system. The same requirement applies to finding the mole fraction of Ar. Without this specific quantitative data, the values cannot be determined.
- Conclusion and Required Information
To find the mole fraction of any gas component \(i\) in the future, one would use the formula:
\[X_i = \frac{P_i}{P_T}\]
where \(P_i\) is the partial pressure of component \(i\) and \(P_T\) is the total pressure. Alternatively, if molar data is available, one would use:
\[X_i = \frac{n_i}{n_T}\]
where \(n_i\) is the number of moles of component \(i\) and \(n_T\) is the total number of moles. Until such data is provided, the mole fraction remains an unknown variable dependent on the missing measurements.
Examples
In a chemistry laboratory, if a researcher collects a sample of air and measures the partial pressures of its components using a sensitive pressure transducer, they can determine the composition of the sample. For example, if the partial pressure of oxygen is 0.21 atm and the total atmospheric pressure is 1.00 atm, the researcher calculates the mole fraction of oxygen as \(X_{O_2} = \frac{0.21 \text{ atm}}{1.00 \text{ atm}} = 0.21\). This dimensionless value directly indicates that 21% of the gas molecules in the sample are oxygen, providing essential data for studying air quality or combustion efficiency.