Problem 64 - Chemistry: The Central Science in SI Units, Global Edition

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 oxygen is \(391.54 \text{ Pa}\), the partial pressure of helium is \(1987.18 \text{ Pa}\), and the total pressure is \(2378.72 \text{ Pa}\).

Step 1: Convert the mass of each gas to moles

We need to determine the amount of each gas in moles. To do this, we will use the molar mass of each gas: Molar masses: Oxygen (O2): 32 g/mol Helium (He): 4 g/mol Number of moles (n) = mass (m) / molar mass Moles of O2: \(n_{O_2} = \frac{51.2 \text{ g}}{32 \text{ g/mol}} = 1.6 \text{ mol}\) Moles of He: \(n_{He} = \frac{32.6 \text{ g}}{4 \text{ g/mol}} = 8.15 \text{ mol}\)

Step 2: Convert temperature to Kelvin

To use the Ideal Gas Law, we need to convert the Celsius temperature given into Kelvin. \(T (K) = T (^\circ C) + 273.15\) \(T = 19^\circ C + 273.15 = 292.15 \text{ K}\)

Step 3: Calculate the ideal gas constant

The ideal gas constant (R) is given in SI units as \(8.314 \frac{\text{J}}{\text{mol~K}}\).

Step 4: Find the partial pressure of each gas using the Ideal Gas Law

The Ideal Gas Law states: \(PV = nRT\), where P is pressure, V is volume, n is the amount of gas in moles, R is the ideal gas constant, and T is temperature in Kelvin We will rearrange the Ideal Gas Law to find the partial pressure of each gas: \(P = \frac{nRT}{V}\) Partial pressure of O2: \(P_{O_2} = \frac{n_{O_2}RT}{V} = \frac{1.6 \text{ mol} \cdot 8.314 \frac{\text{J}}{\text{mol~K}} \cdot 292.15 \text{ K}}{10.0 \text{ L}} = 391.54 \text{ Pa}\) Partial pressure of He: \(P_{He} = \frac{n_{He}RT}{V} = \frac{8.15 \text{ mol} \cdot 8.314 \frac{\text{J}}{\text{mol~K}} \cdot 292.15 \text{ K}}{10.0 \text{ L}} = 1987.18 \text{ Pa}\)

Step 5: Determine the total pressure using Dalton's Law of partial pressures

According to Dalton's Law of partial pressures, the total pressure of a mixture of ideal gases is the sum of the partial pressures of its individual gases: \(P_{total} = P_{O_2} + P_{He}\) \(P_{total} = 391.54 \text{ Pa} + 1987.18 \text{ Pa} = 2378.72 \text{ Pa}\) Therefore, the partial pressure of oxygen is \(391.54 \text{ Pa}\), the partial pressure of helium is \(1987.18 \text{ Pa}\), and the total pressure is \(2378.72 \text{ Pa}\).

Key Concepts

Partial Pressure

Partial pressure refers to the pressure exerted by a single type of gas in a mixture of gases. It can be thought of as the contribution each gas makes to the total pressure within a contained space. This concept is crucial when predicting how gases behave in various chemical reactions and physical conditions. In the case of our exercise, we have the gases oxygen and helium mixed within a cylinder.

• Each type of gas exerts its own pressure, known as its partial pressure, which is independent of the other gases present.
• The ideal gas law, given by the equation \( PV = nRT \), is used to calculate this pressure for each gas separately.

To determine the partial pressures of oxygen and helium, we use the formula \( P = \frac{nRT}{V} \) for each gas.
Thus, by knowing the moles \( n \) of the gases, the temperature \( T \) in Kelvin, and the volume \( V \), we can find each gas's partial pressure within the cylinder.

Dalton's Law of Partial Pressures

Dalton's Law of Partial Pressures is a fundamental principle in gas behavior. It states that the total pressure exerted by a mixture of non-reacting gases can be calculated as the sum of the partial pressures of each individual gas component in the mixture. This law helps us understand the collective behavior of gases when combined.

• The formula is \( P_{\text{total}} = P_1 + P_2 + P_3 + \ldots \), where each \( P \) represents the partial pressure of an individual gas in the mixture.
• For our exercise, the total pressure is the sum of the partial pressures of oxygen and helium.

Using Dalton's Law and our calculated partial pressures from the ideal gas law, we sum these pressures: \( P_{\text{total}} = 391.54 \text{ Pa} + 1987.18 \text{ Pa} = 2378.72 \text{ Pa} \). This approach simplifies calculating the overall pressure in a gas mixture.

Molar Mass

Molar mass is an important concept when dealing with chemical quantities and reactions. It relates the mass of a substance to the amount in moles, a standard unit used in chemistry to quantify particles.

• Molar mass helps us convert a given mass of a substance to the number of moles, using the formula: \( n = \frac{m}{\text{molar mass}} \).
• In our example, the molar mass of oxygen (\( O_2 \)) is 32 g/mol, while helium (He) has a molar mass of 4 g/mol.

Knowing the masses of oxygen and helium in the mixture, we calculated the number of moles of each gas. This step is pivotal for using the ideal gas law to find the gases' partial pressures.

Conversion to Kelvin

Temperature plays a vital role in calculations involving gases, such as those using the ideal gas law. Since this law requires temperature in Kelvin, it is often necessary to convert from Celsius.

• The conversion formula is \( T(K) = T(°C) + 273.15 \).
• This conversion is crucial because Kelvin is an absolute temperature scale, which means it starts at absolute zero, the lowest possible temperature where particles have minimal kinetic energy.

In the problem, the given temperature is \( 19°C \). Converting to Kelvin, we get \( 19 + 273.15 = 292.15 \text{ K} \). Using Kelvin ensures accuracy when applying the ideal gas law to determine the behavior of gases under specific conditions.