Problem 63 - Chemistry The Central Science In Si Units

Question

A mixture containing \(0.50 \mathrm{~mol} \mathrm{H}_{2}(g), 1.00 \mathrm{~mol} \mathrm{O}_{2}(g)\), and 3.50 \(\mathrm{mol} \mathrm{N}_{2}(g)\) is confined in a 25.0-L vessel at \(25^{\circ} \mathrm{C}\). (a) Calculate the total pressure of the mixture. (b) Calculate the partial pressure of each of the gases in the mixture.

Step-by-Step Solution

Answer

Total pressure is approximately 4.91 atm. Partial pressures: \(P_{H_2} \approx 0.491\) atm, \(P_{O_2} \approx 0.982\) atm, \(P_{N_2} \approx 3.437\) atm.

Step 1: Understand the Ideal Gas Law

The ideal gas law equation is given by \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant \( 0.0821 \frac{L \, atm}{mol \, K} \), and \( T \) is the temperature in Kelvin.

Step 2: Convert Temperature to Kelvin

Given the temperature is \(25^{ ext{o}}C\), convert it to Kelvin using the formula: \[ T(K) = T(^{ ext{o}}C) + 273.15 \] Substitute \(25^{ ext{o}}C\): \[ T = 25 + 273.15 = 298.15 \, K \]

Step 3: Calculate Total Moles of Gas

Add the moles of each gas given: - \(0.50\) mol of \(H_2\)- \(1.00\) mol of \(O_2\)- \(3.50\) mol of \(N_2\)Total moles, \( n_{total} = 0.50 + 1.00 + 3.50 = 5.00 \, mol \)

Step 4: Calculate Total Pressure

Use the ideal gas law to find the total pressure:\[ P_{total} = \frac{n_{total}RT}{V} \] Substitute \( n_{total} = 5.00 \) mol, \( R = 0.0821 \frac{L \, atm}{mol \, K} \), \( T = 298.15 \, K \), and \( V = 25.0 \, L \):\[ P_{total} = \frac{5.00 \times 0.0821 \times 298.15}{25.0} \approx 4.91 \, atm \]

Step 5: Calculate Partial Pressures

Partial pressure of each gas is calculated using the formula for each gas: \[ P_i = \left(\frac{n_i}{n_{total}}\right) \times P_{total} \]- \( H_2: P_{H_2} = \left(\frac{0.50}{5.00}\right) \times 4.91 \approx 0.491 \, atm \) - \( O_2: P_{O_2} = \left(\frac{1.00}{5.00}\right) \times 4.91 \approx 0.982 \, atm \) - \( N_2: P_{N_2} = \left(\frac{3.50}{5.00}\right) \times 4.91 \approx 3.437 \, atm \)

Key Concepts

Partial Pressure

In any mixture of gases, each component exerts its own pressure, called partial pressure. It's a way to express the contribution of individual gases to the total pressure in the system. The concept of partial pressure stems from Dalton's Law of Partial Pressures. This law states that the total pressure of a gas mixture is the sum of the partial pressures of all the individual gases in the mixture.

To calculate the partial pressure of a specific gas, you can use the formula:

• \( P_i = \left(\frac{n_i}{n_{total}}\right) \times P_{total} \)

Where:

• \( P_i \) = partial pressure of the gas
• \( n_i \) = number of moles of the gas
• \( n_{total} \) = total number of moles in the mixture
• \( P_{total} \) = total pressure of the mixture

By multiplying the fraction of the total moles that a particular gas makes up by the total pressure, you find the pressure that particular gas contributes to the system. This calculation allows us to better understand the behavior and interaction of gases in a mixture.

Molar Concentration

Molar concentration refers to the amount of a substance (in moles) present in a given volume of solution or mixture. In the context of gases, it's akin to the concentration of each gas in a gas mixture. It helps us understand how much of a particular gas is available in a specified volume.

While molar concentration is a term often used in solutions, it's applicable to gases in terms of calculating mole fractions or partial pressures. It is often expressed in moles per liter (mol/L) and can be calculated using:

• \( c = \frac{n}{V} \)

Where:

• \( c \) = molar concentration
• \( n \) = number of moles of gas
• \( V \) = volume of the container holding the gas

In gas mixtures, understanding the molar concentration of each component helps determine how they interact and react with each other. It is essential for calculating partial pressures and understanding the overall behavior of gases under varying conditions.

Gas Mixtures

A gas mixture consists of two or more different gases that are combined in a single container. Unlike solids and liquids, gases do not separate out based on density in a mixture, but rather, they spread evenly throughout the available space. This property is due to the kinetic theory of gases, which suggests that gas particles are in constant, random motion.

In a gas mixture, each component gas behaves independently of the others, as if it were the only gas present, according to the ideal gas law. This independent behavior is what allows us to calculate partial pressures for each component. Gas mixtures can be found in several practical applications such as air, natural gas, and industrial processes.

• They are frequently analyzed to determine the concentration of individual components.
• This information is used in quality control, environmental testing, and chemical synthesis.

Understanding gas mixtures enables us to harness their properties effectively and predict their behavior under different conditions.

Temperature Conversion

Temperature conversion is a crucial aspect when working with gases, as temperature must be addressed through consistent units for accurate calculations. The Ideal Gas Law, for example, requires temperature in the Kelvin scale. This is because Kelvin is an absolute temperature scale, meaning it starts from absolute zero—the lowest theoretical temperature where particles have minimal kinetic energy.

To convert Celsius to Kelvin, use the formula:

• \( T(K) = T(^{\circ}C) + 273.15 \)

This conversion is necessary due to the direct relationship between temperature and other gas properties like pressure and volume. Neglecting to convert the temperature can lead to errors, as the Kelvin scale ensures these relationships are proportional and meaningful.

• Without Kelvin, calculations involving gas laws can become inaccurate because the scales would not be absolute.
• Temperature conversion thus ensures consistency and accuracy in calculations and scientific reporting.