Question

When working with a mixture of gases, it is sometimes convenient to use an apparent molar mass (a weightedaverage molar mass). Think in terms of replacing the mixture with a hypothetical single gas. What is the apparent molar mass of air, given that air is \(78.08 \% \mathrm{N}_{2}\) \(20.95 \% \mathrm{O}_{2,0.93 \%} \mathrm{Ar}_{\left(\text {and } 0.036 \% \mathrm{CO}_{2},\text { by volume? } \right.}\)

Short answer

The apparent molar mass of air is the sum of each component's molar mass times its fraction in the mixture. This is calculated as \((28 g/mol \times 0.7808) + (32 g/mol \times 0.2095) + (40 g/mol \times 0.0093) + (44 g/mol \times 0.00036)\), giving you the apparent molar mass of air in g/mol. Calculate this operation to find the numeric answer.

Step-by-step solution

Identify the molar masses of the individual gases

Refer to the periodic table or other reliable source to find the molar masses of N2 (28 g/mol), O2 (32 g/mol), Ar (40 g/mol), and CO2 (44 g/mol).

Convert the percentage composition to a decimal

\(\mathrm{N}_{2}\) makes up 78.08% of air, which is 0.7808 in decimal form. Similarly, \(\mathrm{O}_{2}\) is 20.95% = 0.2095, \(\mathrm{Ar}\) is 0.93% = 0.0093 and \(\mathrm{CO}_{2}\) is 0.036% = 0.00036.

Use the percentage composition to find the contribution to the apparent molar mass from each component

For each gas, multiply its molar mass by its percentage (in decimal form) in the mixture. For N2, it will be \(28 \times 0.7808\) g/mol. Repeat the process for each gas.

Sum up the contributions to find the total apparent molar mass

After calculating contributions for each component, sum them to get the final apparent molar mass of air.

Key concepts

Gas Mixtures

In everyday life, we often encounter situations where different gases mix together. Air is a classic example of a gas mixture, comprising primarily nitrogen (\(\mathrm{N}_2\)), oxygen (\(\mathrm{O}_2\)), argon (\(\mathrm{Ar}\)), and carbon dioxide (\(\mathrm{CO}_2\)). When gases mix, they do so without reacting chemically, which makes a gas mixture an interesting component to study.

In a mixture, each gas occupies the same volume and contributes to the overall properties of the gas. However, measuring individual amounts can be cumbersome, hence the use of an "apparent molar mass." This single number represents a weighted average of all the gaseous components, simplifying calculations.

Understanding gas mixtures allows us to practically apply the concept of apparent molar mass, which is useful in diverse scientific fields such as atmospheric science, where knowing the exact composition of air can be critical.

Molar Masses

Molar mass is an essential concept in chemistry that helps us understand how chemical substances combine and interact. It is a measure of the mass of one mole of a substance, usually expressed in grams per mole (g/mol). For gases, it is crucial in defining characteristics such as density and the behavior when mixed.

For example, nitrogen (\(\mathrm{N}_2\)) has a molar mass of 28 g/mol, while oxygen (\(\mathrm{O}_2\)) is 32 g/mol. Argon (\(\mathrm{Ar}\)) and carbon dioxide (\(\mathrm{CO}_2\)) are slightly heavier at 40 g/mol and 44 g/mol, respectively. Understanding these values helps us calculate the apparent molar mass of a gas mixture like air.

When assessing a gas mixture, each component's molar mass contributes based on its volume percentage. It is essential to multiply each gas's molar mass by its fractional volume in the mix, which you then sum up. This calculation gives the apparent molar mass, a handy figure for theoretical and practical applications.

Percentage Composition

The percentage composition of a gas mixture refers to the proportion of each component gas within a mixture, expressed as a percentage. This is a key aspect when determining the role and presence of each gas in the mixture, particularly when calculating apparent molar mass.

Each component gas has a specific percentage that must be converted into a decimal to use in mathematical calculations. For example, air consists of \(78.08\%\) nitrogen, \(20.95\%\) oxygen, \(0.93\%\) argon, and \(0.036\%\) carbon dioxide. These percentages translate to decimal forms: \(0.7808\) for nitrogen, \(0.2095\) for oxygen, \(0.0093\) for argon, and \(0.00036\) for carbon dioxide.

Once in decimal form, these figures can be multiplied by the respective molar masses to find each component's contribution to the total apparent molar mass. This systematic approach simplifies the computation and offers a streamlined way to understand the gaseous mixture's overall nature.