Question

For scuba dives below 150 \(\mathrm{ft}\) , helium is often used to replace nitrogen in the scuba tank. If 15.2 \(\mathrm{g}\) of \(\mathrm{He}(g)\) and 30.6 \(\mathrm{g}\) of \(\mathrm{O}_{2}(g)\) are added to a previously evacuated 5.00 \(\mathrm{L}\) tank at \(22^{\circ} \mathrm{C},\) calculate the partial pressure of each gas present as well as the total pressure in the tank.

Short answer

The partial pressure of helium is 18.3 atm, the partial pressure of oxygen is 4.71 atm, and the total pressure in the tank is 23.0 atm.

Step-by-step solution

Calculate the number of moles of each gas

To find the number of moles of each gas, use the molar mass. The molar mass of He is 4.00 g/mol and of O₂ is 32.00 g/mol. For helium: \(n_{He} = \frac{15.2 \,\text{g}}{4.00\, \text{g/mol}} = 3.80\, \text{mol}\) For oxygen: \(n_{O_2} = \frac{30.6\, \text{g}}{32.00\, \text{g/mol}} = 0.956\, \text{mol}\)

Convert temperature to Kelvin

To use the Ideal Gas Law, we need to convert the temperature from Celsius to Kelvin. \(T_K = 22^{\circ} \text{C} + 273.15 = 295.15\,\text{K}\)

Apply the Ideal Gas Law for each gas

Now we will use the Ideal Gas Law formula (PV=nRT) separately for both helium and oxygen, to calculate their partial pressures. For helium: \(P_{He}V= n_{He}RT\) \(P_{He} = \frac{n_{He}RT}{V} = \frac{3.80\,\text{mol}\cdot 0.0821\,\frac{\text{L}\cdot \text{atm}}{\text{mol}\cdot \text{K}}\cdot 295.15\,\text{K}}{5.00\,\text{L}}= 18.3\,\text{atm}\) For oxygen: \(P_{O_2}V= n_{O_2}RT\) \(P_{O_2} = \frac{n_{O_2}RT}{V} = \frac{0.956\,\text{mol}\cdot 0.0821\,\frac{\text{L}\cdot \text{atm}}{\text{mol}\cdot \text{K}}\cdot 295.15\,\text{K}}{5.00\,\text{L}}= 4.71\,\text{atm}\)

Calculate the total pressure

The total pressure in the tank is the sum of the partial pressures of helium and oxygen. Total pressure: \(P_{total} = P_{He} + P_{O_2} = 18.3\,\text{atm} + 4.71\,\text{atm} = 23.0\,\text{atm}\) So, the partial pressure of helium is 18.3 atm, the partial pressure of oxygen is 4.71 atm, and the total pressure in the tank is 23.0 atm.

Key concepts

Partial Pressure

Partial pressure is an essential concept when working with gas mixtures. In a mixture of gases, each gas exerts pressure independently as if it were the only gas present. This is known as its partial pressure. Think of it as each gas having its own little "spotlight". The total pressure in a container is the sum of the partial pressures of all the gases inside it. This is beautifully illustrated in Dalton's Law of Partial Pressures:

• The partial pressure of a gas is the product of its mole fraction and the total pressure.
• The total pressure in a gas mixture can be calculated as the sum of all individual partial pressures: \( P_{total} = P_{gas1} + P_{gas2} + ... + P_{gasn} \).

For instance, in a scuba tank with helium and oxygen, knowing the partial pressures helps us understand the composition of breathable air within, ensuring it is safe for deep-sea diving.

Mole Calculation

Calculating the number of moles is crucial for working with gases, especially when using the Ideal Gas Law. The number of moles reflects how many molecules there are of a substance. We use the formula:\[ n = \frac{\text{mass in grams}}{\text{molar mass in g/mol}}\]Here’s how:

• Molar Mass: Helium has a molar mass of 4.00 g/mol, while oxygen is 32.00 g/mol.
• Convert: For example, 15.2 g of helium gives \(3.80 \text{ mol of } \text{He}\).
• Consistency: Ensure mass is in grams and molar mass is accurate for accurate mole determination.

Understanding mole calculation is foundational for determining how much gas is present, which is critical for calculating partial pressures and performing meaningful experiments.

Temperature Conversion

Temperature conversion from Celsius to Kelvin is a small but vital step when working with gas laws. Kelvin is the preferred unit in scientific calculations because it’s an absolute temperature scale.Why convert to Kelvin?

• The Kelvin scale starts at absolute zero, ensuring no negative values which could complicate gas law calculations.
• Conversion is straightforward: \( T_K = T_\text{C} + 273.15 \), so for our example, \( 22^\circ \text{C} = 295.15 \text{ K}\).

Remembering to use Kelvin ensures that all variables in the Ideal Gas Law, \( PV = nRT \), remain consistent and the calculations are valid.

Gas Mixtures

Gas mixtures, like those in scuba tanks, often include multiple gases combined to achieve the desired properties. For scuba divers, breathing a mixture that substitutes helium for nitrogen reduces the risk of nitrogen narcosis. Understanding gas mixtures involves:

• Composition: Each gas in the mixture contributes to the total pressure and the properties of the mixture.
• Individual Behavior: Even when mixed, gases behave independently, adhering to their physical laws like the Ideal Gas Law.
• Safety: Different gas ratios suit different depths and conditions, making comprehension crucial of things like buoyancy and narcosis.

Employing effective gas mixtures is not just about solving chemistry problems but ensuring the safety and success of activities like deep-sea diving.