Question

Calculate the total pressure (in atm) of a mixture of \(0.0300 \mathrm{~mol}\) of helium, He, and \(0.0400 \mathrm{~mol}\) of oxygen, \(\mathrm{O}_{2}\) in a 4.00-L flask at \(20^{\circ} \mathrm{C}\). Assume ideal gas behavior.

Short answer

The total pressure of the gas mixture is 0.421 atm.

Step-by-step solution

Identify the Known Variables

We are given the number of moles of helium (\(n_{He} = 0.0300 \text{ mol}\)), the number of moles of oxygen (\(n_{O_2} = 0.0400 \text{ mol}\)), the volume of the container (\(V = 4.00 \text{ L}\)), and the temperature (\(T = 20^{\circ} \text{C} = 293 \text{ K}\)). We also need the universal gas constant \(R = 0.0821 \text{ L}\cdot \text{atm}/(\text{mol}\cdot \text{K})\).

Calculate Partial Pressure of Helium

Use the ideal gas law, \(PV = nRT\), to find the partial pressure of helium. Substituting in the known values for helium:\[ P_{He} = \frac{n_{He}RT}{V} = \frac{0.0300 \times 0.0821 \times 293}{4.00} = 0.180 \text{ atm} \]

Calculate Partial Pressure of Oxygen

Similarly, use the ideal gas law to find the partial pressure of oxygen. Substituting the known values for oxygen:\[ P_{O_2} = \frac{n_{O_2}RT}{V} = \frac{0.0400 \times 0.0821 \times 293}{4.00} = 0.241 \text{ atm} \]

Calculate Total Pressure

The total pressure is the sum of the partial pressures of helium and oxygen due to Dalton's Law of Partial Pressures. Thus:\[ P_{total} = P_{He} + P_{O_2} = 0.180 + 0.241 = 0.421 \text{ atm} \]

Key concepts

Partial Pressure

Partial pressure is a concept used in chemistry to describe the pressure exerted by a specific gas within a mixture of gases. Each gas in a mixture, even though they are mixed together, behaves independently of the others. This concept is crucial when dealing with environments where multiple gases are present, such as inside the human lungs or in a chemical reaction involving gas mixtures.

To calculate the partial pressure of a gas, we use the Ideal Gas Law, which is expressed as:

• P (pressure)
• V (volume)
• n (number of moles)
• R (universal gas constant)
• T (temperature in Kelvin)

The formula is given by: \[ P = \frac{nRT}{V} \]Where

• P is the partial pressure we want to calculate
• n is the number of moles of the specific gas

Using this formula, you can determine the partial pressure contributed by each gas in the mixture.

Dalton's Law of Partial Pressures

Dalton's Law of Partial Pressures is a fundamental principle in gases, particularly when dealing with mixtures. This law states that the total pressure exerted by a mixture of gases is equal to the sum of the partial pressures of each individual gas in the mixture.

Mathematically, this is expressed as:

• \[P_{total} = P_1 + P_2 + P_3 + ...\]

Where

• P_total is the total pressure of the mixture
• P₁, P₂, P₃, ... are the partial pressures of the individual gases

This law is particularly useful in scenarios like the mixture of helium and oxygen in a flask, where the total pressure is needed. By individually calculating the partial pressures of each gas, as shown in the Ideal Gas Law calculations, we can simply add them up to find the total pressure. This is why understanding both the concept of partial pressure and Dalton's Law is essential.

Universal Gas Constant

The Universal Gas Constant, denoted by the symbol R, is a key term in the Ideal Gas Law equation, linking the properties of pressure, volume, and temperature for ideal gases. The value of R is determined to be 0.0821 L·atm/(mol·K) when using pressure in atmospheres and volume in liters.

This constant is crucial as it serves as a bridge in calculations involving gases, enabling us to quantify the relationship between moles and pressure under ideal conditions. In real-world applications, knowing R helps you perform accurate calculations of gas behavior where the Ideal Gas Law is applicable.

Understanding the Universal Gas Constant helps in a wide range of applications:

• Predicting the behavior of gases at different temperatures or pressures
• Determining how gases will react in chemical reactions

Thus, mastering the use of this constant is essential for anyone studying or working with the properties of gases.