Problem 81 - Chemistry The Molecular Nature of Matter

Question

A mixture of gases contains 315 torr \(\mathrm{N}_{2}, 275\) torr \(\mathrm{O}_{2}\), and 285 torr Ar. What is the total pressure of the mixture? What is the mole fraction of \(\mathrm{O}_{2}\) in this mixture?

Step-by-Step Solution

Answer

The total pressure of the gas mixture is 875 torr, and the mole fraction of \(\mathrm{O}_2\) is approximately 0.3143.

Step 1: Calculating the Total Pressure

The total pressure of a mixture of gases can be found by adding together the partial pressures of each individual gas in the mixture. Use Dalton's Law of Partial Pressures which states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of each component gas. In this case, add the pressure contributions from \(\mathrm{N}_2\), \(\mathrm{O}_2\), and Ar.

Step 2: Finding the Mole Fraction

To find the mole fraction of \(\mathrm{O}_2\), use the formula for mole fraction, which is the partial pressure of the gas divided by the total pressure. The mole fraction is a way of expressing the proportion of a gas in a mixture, with reference to the number of moles of gas.

Step 3: Final Values

Report the calculated total pressure of the gas mixture. Then, report the mole fraction of \(\mathrm{O}_2\) as the final value, rounded appropriately if necessary.

Key Concepts

Dalton's Law of Partial Pressures

Understanding Dalton's Law of Partial Pressures is central to calculating the total pressure of a gas mixture. Generally attributed to the English chemist John Dalton, this law reveals how a mixture of non-reacting gases behaves under pressure. Each gas in a blend contributes to the total pressure independently, as if it were alone in the container.

This individual pressure contribution is known as the gas's partial pressure. Thus, the sum of all partial pressures gives us the total pressure. The importance of this law lies in its simplicity and broad application across various scientific problems and practical scenarios, such as calculating atmospheric pressure or designing chemical processes. When dealing with mixtures, it's a reliable starting point for analysis.

Mole Fraction

The mole fraction is a dimensionless number that provides a measure of the concentration of a particular component in a mixture. It is defined as the ratio of the number of moles of that component to the total number of moles of all components in the mixture. What's noteworthy about mole fractions is that they sum up to 1 when considering all components of the mixture.

For instance, if we know the mole fraction of oxygen in a container, we know what percentage of the total moles is made up of oxygen. This number can be crucial when preparing chemical reactions or when analyzing the composition of a mixture. It's commonly used because it's independent of temperature and pressure, which makes it a very versatile unit in gas calculations and chemical engineering.

Partial Pressure

The concept of partial pressure is a cornerstone for understanding gaseous systems. It's the pressure that a gas would exert if it occupied the entire volume of the mixture at the same temperature. The term 'partial' signifies that it is the portion of the total pressure attributable to that specific gas.

When working with partial pressures, remember that they are directly proportional to the number of moles of gas (provided the temperature and volume remain consistent). Therefore, if you know the total pressure and the mole fraction, you can determine the partial pressure of any component in the mixture. Similarly, given the partial pressure and the total pressure, you can backtrack to find a component's mole fraction. The ability to switch seamlessly between these concepts allows for greater flexibility in handling gas-related calculations.