Question

The partial pressure of \(\mathrm{CH}_{4}(g)\) is \(0.175 \mathrm{~atm}\) and that of \(\mathrm{O}_{2}(g)\) is \(0.250\) atm in a mixture of the two gases. a. What is the mole fraction of each gas in the mixture? b. If the mixture occupies a volume of \(10.5 \mathrm{~L}\) at \(65^{\circ} \mathrm{C}\), calculate the total number of moles of gas in the mixture. c. Calculate the number of grams of each gas in the mixture.

Short answer

a. Calculate the mole fraction of each gas: Step 1: Total pressure, \(P_{total} = 0.175 + 0.250 = 0.425\) atm Step 2: Mole fractions: \(X_{CH4} = \frac{0.175}{0.425} = 0.412\) and \(X_{O2} = \frac{0.250}{0.425} = 0.588\) b. Calculate the total number of moles of gas in the mixture: Step 3: \(T (K) = 65 ^\circ C + 273.15 = 338.15 K\) Step 4: \(n = \frac{PV}{RT} = \frac{(0.425 \mathrm{~atm})(10.5 \mathrm{~L})}{(0.0821 \frac{\mathrm{L\cdot atm}}{\mathrm{mol\cdot K}})(338.15 \mathrm{~K})} = 0.179\) mol c. Calculate the number of grams of each gas in the mixture: Step 5: Moles of CH4 = \(0.412 * 0.179 = 0.0737\) mol Moles of O2 = \(0.588 * 0.179 = 0.105\) mol Step 6: Mass of CH4 (grams) = \(0.0737\,\mathrm{mol} * 16.04 \frac{\mathrm{g}}{\mathrm{mol}} = 1.18 \mathrm{~g}\) Mass of O2 (grams) = \(0.105\,\mathrm{mol} * 32.00 \frac{\mathrm{g}}{\mathrm{mol}} = 3.36 \mathrm{~g}\)

Step-by-step solution

Calculate the total pressure

Add the partial pressures of each gas to find the total pressure of the gas mixture: \(P_{total} = P_{CH_4} + P_{O_2}\)

Calculate the mole fraction of each gas

Divide the pressure of each gas by the total pressure: Mole fraction of CH4 (X_{CH4}) = \(\frac{P_{CH_4}}{P_{total}}\) Mole fraction of O2 (X_{O2}) = \(\frac{P_{O_2}}{P_{total}}\) b. Calculate the total number of moles of gas in the mixture:

Convert the temperature to Kelvin

In the Ideal Gas Law, the temperature must be in Kelvin, so convert the given temperature from Celsius to Kelvin: T (K) = 65 °C + 273.15

Use Ideal Gas Law to find total moles

Rearrange the Ideal Gas Law to solve for total moles (n): n = \(\frac{PV}{RT}\) Since we have P_total, V, R (gas constant), and T, we can find the total number of moles (n). c. Calculate the number of grams of each gas in the mixture:

Calculate moles of each gas using mole fraction

Use the mole fraction and total moles to find the moles of each gas: Moles of CH4 = X_{CH4} * total moles Moles of O2 = X_{O2} * total moles

Convert moles to grams using molar mass

Use the molar mass of each gas to convert the moles to grams: Molar mass of CH4 = 16.04 g/mol Molar mass of O2 = 32.00 g/mol Miles of CH4 (grams) = Moles of CH4 * Molar mass of CH4 Miles of O2 (grams) = Moles of O2 * Molar mass of O2

Key concepts

Partial Pressure

Partial pressure is a concept that helps us understand how individual gases behave when mixed together. In a mixture of gases, each gas exerts pressure on the walls of the container. This is called partial pressure. It represents the contribution of each gas to the total pressure, which is the sum of all individual partial pressures.

• For example, in the exercise, the partial pressure of methane (\( \mathrm{CH}_4 \)) is 0.175 atm, and for oxygen (\( \mathrm{O}_2 \)), it is 0.250 atm.
• The total pressure is simply the sum of these two partial pressures, which helps us understand the behavior and interaction of gases.

Partial pressure is directly related to the concept of mole fraction, as the ratio of a gas's partial pressure to the total pressure gives us the mole fraction of that gas.

Mole Fraction

The mole fraction is a way to express the concentration of a gas in a mixture. It's a dimensionless number that represents the ratio of the number of moles of one component to the total number of moles in the mixture.

To calculate the mole fraction:

• Divide the partial pressure of the gas of interest by the total pressure of all gases in the mixture.
• For example, the mole fraction of methane (\( X_{\mathrm{CH}_4} \)) is calculated as \( \frac{P_{CH_4}}{P_{total}} \).

This tells us what fraction of the total pressure is due to each component, making it a handy way to compare different gases in the mixture. Knowing the mole fraction is crucial for finding the number of moles present and further calculations like converting moles to mass.

Molar Mass

Molar mass is the weight of one mole of a substance, given in grams per mole (g/mol). It represents the mass of the Avogadro's number of molecules or atoms in that substance. Each element and compound has a unique molar mass, which we can use to convert between mass and moles.

For example:

• Methane (\(\mathrm{CH}_4\)) has a molar mass of 16.04 g/mol.
• Oxygen (\(\mathrm{O}_2\)) has a molar mass of 32.00 g/mol.

In the exercise, once we find the moles of each gas using the mole fraction, the molar mass helps us convert these amounts to grams. This allows for easy quantification of how much gas is actually present in terms of mass, which is often more practical for experimental purposes.

Temperature Conversion

Temperature conversion is an important step when using the Ideal Gas Law because this law requires temperature to be in Kelvin (K), not in Celsius (°C). The Kelvin scale starts from absolute zero, making it a standard for scientific calculations.

To convert Celsius to Kelvin:

• Add 273.15 to your Celsius temperature.
• For instance, if the temperature is 65°C in the exercise, convert it as follows: \( T(K) = 65 + 273.15 = 338.15 \) K.

Converting the temperature ensures accuracy in calculations involving volume, pressure, and the Ideal Gas Law equation \( n = \frac{PV}{RT} \). With the temperature correctly converted to Kelvin, you can determine the total number of moles of gases in the mixture accurately.