Question

A mixture consisting of \(7.0 \mathrm{~g}\) of \(\mathrm{CO}\) and \(10.0 \mathrm{~g}\) of \(\mathrm{SO}_{2},\) two atmospheric pollutants, has a pressure of 0.33 atm when placed in a sealed container. What is the partial pressure of CO?

Short answer

The partial pressure of \(\text{CO}\) is \0.203\text{ atm}\.

Step-by-step solution

- Calculate moles of \(\text{CO}\)

Use the molar mass of \(\text{CO}\) to convert grams to moles. The molar mass of \(\text{CO}\) is 28.01 g/mol. \Number of moles of \(\text{CO} = \frac{7.0 \text{ g}}{28.01 \text{ g/mol}} = 0.25 \text{ mol} \).

- Calculate moles of \(\text{SO}_{2}\)

Similarly, use the molar mass of \(\text{SO}_{2}\) to convert grams to moles. The molar mass of \(\text{SO}_{2}\) is 64.07 g/mol. \Number of moles of \(\text{SO}_{2} = \frac{10.0 \text{ g}}{64.07 \text{ g/mol}} = 0.156 \text{ mol} \).

- Total moles in the mixture

Add the moles of \(\text{CO}\) and \(\text{SO}_{2}\) to find the total moles in the mixture. \Total moles \(\text{n}_{\text{total}} = 0.25 \text{ mol} + 0.156 \text{ mol} = 0.406 \text{ mol} \).

- Calculate mole fraction of \(\text{CO}\)

The mole fraction \(\text{X}_{\text{CO}}\) is the ratio of moles of \(\text{CO}\) to the total moles. \Mole fraction \(\text{X}_{\text{CO}} = \frac{0.25 \text{ mol}}{0.406 \text{ mol}} = 0.616\).

- Calculate partial pressure of \(\text{CO}\)

The partial pressure of \(\text{CO}\) is found by multiplying its mole fraction by the total pressure. \Partial pressure of \(\text{CO} = \text{X}_{\text{CO}} \times P_{\text{total}} = 0.616 \times 0.33 \text{ atm} = 0.203 \text{ atm} \).

Key concepts

Molar Mass

Molar mass is an important concept in chemistry, particularly when dealing with gas calculations like partial pressure. It represents the weight of one mole of a given substance. The unit for molar mass is grams per mole (g/mol). To find the molar mass, you sum the atomic masses of all the atoms in a molecule.
Molar mass gives you a way to convert between grams and moles. For example, the molar mass of carbon monoxide (CO) is calculated by adding the atomic masses of carbon (12.01 g/mol) and oxygen (16.00 g/mol) to get 28.01 g/mol.
Knowing how to calculate and use molar mass is crucial for converting grams to moles, which is needed for calculating the number of molecules involved in a reaction or process. This understanding sets the foundation for further calculations involving mole fraction and partial pressure.

Mole Fraction

Mole fraction is a way of expressing the concentration of a component in a mixture. It is the ratio of the number of moles of one component to the total number of moles in the mixture.
To find the mole fraction \(X_{\text{CO}}\) of carbon monoxide (CO) in a mixture, follow this formula:
\[ X_{\text{CO}} = \frac{\text{moles of CO}}{\text{total moles}} \]
In our example, if you have 0.25 moles of CO and the total moles in the mixture is 0.406, then the mole fraction is:
\[ X_{\text{CO}} = \frac{0.25}{0.406} = 0.616 \]
Mole fraction is a dimensionless quantity and is essential in calculating partial pressures of gases in a mixture.

Partial Pressure

Partial pressure describes the pressure exerted by a single type of gas in a mixture of gases. According to Dalton's Law, the total pressure exerted by a gas mixture is equal to the sum of the partial pressures of each individual gas.
To calculate the partial pressure of carbon monoxide (CO) in our example, we use its mole fraction and the total pressure of the gas mixture. The formula is:
\[ P_{\text{CO}} = X_{\text{CO}} \times P_{\text{total}} \]
Given that the mole fraction of CO is 0.616 and the total pressure is 0.33 atm, the partial pressure of CO becomes:
\[ P_{\text{CO}} = 0.616 \times 0.33 \text{ atm} = 0.203 \text{ atm} \]
Partial pressures are useful in many chemical processes and reactions, particularly those involving gases.