Problem 38 - Chemistry and Chemical Reactivity

Question

A cylinder of compressed gas is labeled "Composition (mole \(\%): 4.5 \% \mathrm{H}_{2} \mathrm{S}, 3.0 \% \mathrm{CO}_{2},\) balance \(\mathrm{N}_{2} .\) "The pressure gauge attached to the cylinder reads 46 atm. Calculate the partial pressure of each gas, in atmospheres, in the cylinder.

Step-by-Step Solution

Answer

Partial pressures: \(\text{H}_2\text{S}\) = 2.07 atm, \(\text{CO}_2\) = 1.38 atm, \(\text{N}_2\) = 42.55 atm.

Step 1: Understand Total Pressure

The total pressure in the cylinder is given as 46 atm. We will use this to calculate the partial pressures of each component gas using their mole percentages.

Step 2: Calculate the Fraction of Each Gas

Convert the mole percentages given into fractions of the total. For \(\text{H}_2\text{S}\), it is \(\frac{4.5}{100}\), for \(\text{CO}_2\), it is \(\frac{3.0}{100}\), and for \(\text{N}_2\), it is \(1 - (0.045 + 0.03)\).

Step 3: Apply Dalton's Law of Partial Pressures

Dalton's Law states that the partial pressure of a gas in a mixture is equal to the mole fraction of the gas times the total pressure. Therefore, calculate the partial pressures for each gas as follows: 1. \(\text{H}_2\text{S}\): \(46 \times 0.045 = 2.07\,\text{atm}\)2. \(\text{CO}_2\): \(46 \times 0.03 = 1.38\,\text{atm}\)3. \(\text{N}_2\): \(46 \times (1 - 0.045 - 0.03) = 42.55\,\text{atm}\)

Step 4: Verify the Results

Ensure that the sum of the partial pressures equals the total pressure. Sum the partial pressures: \(2.07 + 1.38 + 42.55 = 46\,\text{atm}\). This confirms that the calculations are correct.

Key Concepts

Mole Fraction

In a mixture of gases, the mole fraction represents the proportion of a given gas relative to the total number of moles in the mixture. It is a way to describe the composition of the gas mixture without being concerned by the actual grams or volume of each gas. The mole fraction of a gas is calculated by dividing the moles of the specific gas by the total moles of all gases present.

For instance, in the given exercise, the mole fractions are derived from the mole percentages. To convert percentage to fraction, simply divide by 100. So for \(\text{H}_2\text{S}\), it becomes \(\frac{4.5}{100}\), for \(\text{CO}_2\), it is \(\frac{3.0}{100}\), and \(\text{N}_2\) is calculated as the remainder \(1 - (0.045 + 0.03)\). This ensures that their cumulative share equals the whole, or one.

• Mole Fraction Formula: \(\text{Mole Fraction} = \frac{\text{Moles of Component}}{\text{Total Moles}}\)
• Significance: Helps in calculating other properties like partial pressures.

Partial Pressure

Partial pressure refers to the individual pressure exerted by a single type of gas in a mixture of gases. This concept is derived from Dalton's Law, which states that the total pressure of a gas mixture is the sum of the partial pressures of each component gas.

The exercise demonstrates how to calculate these pressures. By multiplying the mole fraction of each gas by the total pressure (given as 46 atm), the partial pressure can be calculated for each gas present in the cylinder.

• \(\text{H}_2\text{S}: 46 \times 0.045 = 2.07\,\text{atm}\)
• \(\text{CO}_2: 46 \times 0.03 = 1.38\,\text{atm}\)
• \(\text{N}_2: 46 \times 0.925 = 42.55\,\text{atm}\)

These calculations show how each gas contributes to the overall pressure.

Mixture of Gases

A mixture of gases consists of two or more different gases occupying the same space without reacting chemically. Such mixtures can be found in various natural and industrial scenarios, like atmospheric air or gas tanks in laboratories.

In the exercise, the mixture consists of three different gases: hydrogen sulfide (\(\text{H}_2\text{S}\)), carbon dioxide (\(\text{CO}_2\)), and nitrogen (\(\text{N}_2\)). Each part of the mixture contributes to the total pressure and has its own partial pressure, which depends on its mole fraction.

• The proportion of each gas is given in mole percentage, which is converted into mole fraction to determine individual partial pressures.
• Unlike chemical compounds, gaseous mixtures maintain the properties of each individual gas.

Gas Laws

Gas laws are fundamental principles that explain the behavior of gases under various conditions of temperature, pressure, and volume. They help in predicting and calculating how gases will react in different environments.

One of the gas laws applied in the exercise is Dalton’s Law of Partial Pressures. It aids in understanding how individual gases in a mixture contribute to the total pressure. This is pivotal in many practical situations like adjusting gas concentrations in industrial processes or even clearing up scuba diving tanks.

• Dalton’s Law asserts that the total pressure is equal to the sum of partial pressures of all component gases.
• Other key gas laws include Boyle’s Law, Charles’s Law, and the Ideal Gas Law.
• These laws form the foundation of our understanding and use of gases in both academic and applied sciences.