Problem 64 - Chemistry The Central Science In Si Units

Question

A deep-sea diver uses a gas cylinder with a volume of \(10.0 \mathrm{~L}\) and a content of \(51.2 \mathrm{~g}\) of \(\mathrm{O}_{2}\) and \(32.6 \mathrm{~g}\) of He. Calculate the partial pressure of each gas and the total pressure if the temperature of the gas is \(19^{\circ} \mathrm{C}\).

Step-by-Step Solution

Answer

The partial pressure of \(\mathrm{O}_2\) is approximately 3.75 atm, \(\mathrm{He}\) is approximately 19.14 atm, and the total pressure is approximately 22.89 atm.

Step 1: Convert Temperature to Kelvin

To work with gases, temperature should be in Kelvin. Convert the temperature from Celsius to Kelvin using the formula: \(T(K) = T(C) + 273.15\). So, \(T = 19 + 273.15 = 292.15\, K\).

Step 2: Calculate Moles of Each Gas

Use the molar mass to convert grams to moles. The molar mass of \(\mathrm{O}_2\) is \(32.00\, g/mol\) and for \(\mathrm{He}\) it's \(4.00\, g/mol\). Calculate moles: \[\text{Moles of } \mathrm{O}_2 = \frac{51.2}{32.00} = 1.60\, mol\] \[\text{Moles of He} = \frac{32.6}{4.00} = 8.15\, mol\]

Step 3: Use Ideal Gas Law to Find Total Pressure

The ideal gas law \(PV = nRT\) will help find total pressure. \(n\) is the total moles (sum of moles of \(\mathrm{O}_2\) and \(\mathrm{He}\)), \(R\) is the gas constant \(0.0821\, L\, atm\, mol^{-1}\, K^{-1}\), \(V = 10.0\, L\), \(T = 292.15\, K\). Calculate total pressure: \[P = \frac{(1.60 + 8.15) \times 0.0821 \times 292.15}{10.0} \approx 22.89\, atm\]

Step 4: Calculate Partial Pressure Using Mole Fraction

Partial pressure of a gas can be found using its mole fraction and the total pressure: \(P_{\text{gas}} = X_{\text{gas}} \times P_{\text{total}}\). Calculate mole fractions: \[X_{\mathrm{O}_2} = \frac{1.60}{1.60 + 8.15} = 0.164\] \[X_{\mathrm{He}} = \frac{8.15}{1.60 + 8.15} = 0.836\] Now calculate partial pressures: \[P_{\mathrm{O}_2} = 0.164 \times 22.89 \approx 3.75\, atm\] \[P_{\mathrm{He}} = 0.836 \times 22.89 \approx 19.14\, atm\]

Key Concepts

Partial Pressure

In a mixture of gases, each individual gas contributes to the total pressure. This contribution is known as the partial pressure. To determine the partial pressure of a gas in a mixture, you can use the formula:

• \( P_{\text{gas}} = X_{\text{gas}} \times P_{\text{total}} \)

where \( P_{\text{gas}} \) is the partial pressure of the gas, \( X_{\text{gas}} \) is the mole fraction, and \( P_{\text{total}} \) is the total pressure of the gas mixture.
Partial pressure is a useful concept in chemistry and physics to predict the behavior of gases in a mixture without considering their identities. It simplifies complex calculations and helps in understanding the contributions each gas makes to the overall pressure.
For example, in our problem, the partial pressures were calculated for oxygen (\( \mathrm{O}_2 \)) and helium (He) using their mole fractions.

Mole Fraction

Mole fraction is a way of expressing the concentration of a component in a mixture of gases. It is defined as the number of moles of a specific gas divided by the total number of moles of all gases present. The formula for mole fraction is:

• \( X_{\text{gas}} = \frac{n_{\text{gas}}}{n_{\text{total}}} \)

where \( n_{\text{gas}} \) is the moles of the gas of interest, and \( n_{\text{total}} \) is the total moles of all gases in the mixture.
Mole fractions are always dimensionless numbers between 0 and 1. They provide an understanding of how much of each gas is present compared to others in the mixture.
In the diving exercise you'll notice the mole fraction provided important insights about how each gas contributed to the total pressure, allowing the calculation of partial pressures for \( \mathrm{O}_2 \) and He.

Temperature Conversion

Temperature conversion is crucial in gas law equations to ensure that all calculations are consistent and accurate. The ideal gas law and other gas-related equations require temperature to be in Kelvin, an absolute temperature scale.
The conversion from Celsius to Kelvin is straightforward, using the equation:

• \( T(K) = T(°C) + 273.15 \)

For example, in the exercise, the temperature conversion was necessary to correctly apply the ideal gas law, converting \(19^{\circ} \text{C}\) to \(292.15 \, K\).
This conversion ensures that all scientists and engineers use the same scale, eliminating discrepancies in calculations that could arise if different temperature scales were used.

Molar Mass

Molar mass is a measure of the mass of one mole of a substance, often expressed in grams per mole (g/mol). It plays a key role in converting between the mass of a gas and the amount in moles, which is vital for applying the ideal gas law.
To calculate moles from a given mass:

• \( n = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} \)

In our example, the molar masses of \( \mathrm{O}_2 \) and He were used to determine their respective moles: 1.60 moles for \( \mathrm{O}_2 \) and 8.15 moles for He.
Understanding molar mass allows you to effectively interconvert between the tangible mass of a gas and the abstract concept of moles, aligning with the requirements of the ideal gas law and ensuring accurate pressure calculations.