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

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

Step 1: Identify given information

The total pressure of the gas mixture in the cylinder is 46 atm. The gas composition is given as 4.5% \(\mathrm{H}_2\mathrm{S}\), 3.0% \(\mathrm{CO}_2\), and the balance is \(\mathrm{N}_2\). Mole percentages can be considered as volume percentages for gases.

Step 2: Determine partial pressure formula

Partial pressure can be calculated using the formula: \( P_i = X_i \cdot P_{\text{total}}\), where \(P_i\) is the partial pressure for each gas, \(X_i\) is the mole fraction of the gas, and \(P_{\text{total}}\) is the total pressure.

Step 3: Convert mole percentages to mole fractions

Since percentages can be directly used as mole fractions for gases in standard conditions, convert each gas's mole percentage to a mole fraction: - \( \mathrm{H}_2\mathrm{S}: \frac{4.5}{100} = 0.045 \) - \( \mathrm{CO}_2: \frac{3.0}{100} = 0.030 \) - \( \mathrm{N}_2: 1 - (0.045 + 0.030) = 0.925 \)

Step 4: Calculate partial pressures of each gas

Using the formula from Step 2, calculate the partial pressure for each gas:- \( \mathrm{H}_2\mathrm{S}: P_{\mathrm{H}_2\mathrm{S}} = 0.045 \times 46 = 2.07 \text{ atm} \) - \( \mathrm{CO}_2: P_{\mathrm{CO}_2} = 0.030 \times 46 = 1.38 \text{ atm} \) - \( \mathrm{N}_2: P_{\mathrm{N}_2} = 0.925 \times 46 = 42.55 \text{ atm} \)

Key Concepts

Partial Pressure

The concept of partial pressure is a foundational topic in gas laws. It represents the pressure exerted by a particular gas in a mixture if it alone occupied the entire volume filled by the mixture.
Partial pressure is crucial because it allows us to understand how each gas behaves in a mixture. It's calculated using the formula \( P_i = X_i \times P_{\text{total}} \), where \( P_i \) is the partial pressure of the gas, \( X_i \) is its mole fraction, and \( P_{\text{total}} \) is the total pressure of the gas mixture.

• For hydrogen sulfide (\( \mathrm{H}_2\mathrm{S} \)), if its mole fraction is 0.045 and total pressure is 46 atm, its partial pressure is \( P_{\mathrm{H}_2\mathrm{S}} = 0.045 \times 46 = 2.07 \text{ atm} \).
• Similarly, for carbon dioxide (\( \mathrm{CO}_2 \)), with a mole fraction of 0.030, its partial pressure is \( P_{\mathrm{CO}_2} = 0.030 \times 46 = 1.38 \text{ atm} \).
• Nitrogen (\( \mathrm{N}_2 \)), with the largest mole fraction of 0.925, has a partial pressure of \( P_{\mathrm{N}_2} = 0.925 \times 46 = 42.55 \text{ atm} \).

Recognizing how partial pressures add up to the total pressure helps in applications such as diving, where understanding the behavior of mixed gases is vital.

Mole Fraction

Mole fraction is a way to express the concentration or proportion of a component in a mixture. It is defined as the ratio of the number of moles of a specific component to the total number of moles of all components in the mixture.
Mole fractions are dimensionless numbers that indicate the relative amount of each gas in the mixture. For a gas mixture under standard conditions, mole fractions can often be directly derived from percentage composition data.

• In the given exercise, \( \mathrm{H}_2\mathrm{S} \) has a mole fraction of \( 0.045 \), calculated as \( \frac{4.5}{100} \).
• \( \mathrm{CO}_2 \) has a mole fraction of \( 0.030 \), which equals \( \frac{3.0}{100} \).
• The balance gas, \( \mathrm{N}_2 \), has a mole fraction calculated by subtracting the sums of \( \mathrm{H}_2\mathrm{S} \) and \( \mathrm{CO}_2 \) mole fractions from 1, resulting in \( 0.925 \).

Understanding mole fractions helps us convert between partial pressures and total pressures, and it is a fundamental principle when dealing with gas mixtures.

Dalton's Law of Partial Pressures

Dalton's Law of Partial Pressures is a key principle in understanding gas behavior in mixtures. It states that the total pressure exerted by a mixture of gases is equal to the sum of the partial pressures of its component gases.
This law springs from the idea that gas molecules act independently, each contributing to the total pressure, as if it were alone in the entire volume. Therefore, the equation representing Dalton's Law is: \[ P_{\text{total}} = P_{1} + P_{2} + P_{3} + \ldots \]
In the exercise, this concept is used to determine the total pressure from the calculated partial pressures:

• The partial pressure of \( \mathrm{H}_2\mathrm{S} = 2.07 \text{ atm} \).
• The partial pressure of \( \mathrm{CO}_2 = 1.38 \text{ atm} \).
• The partial pressure of \( \mathrm{N}_2 = 42.55 \text{ atm} \).

These sum to give the total pressure of 46 atm, confirming Dalton's Law. Recognizing this helps in various practical applications, including predicting real-world scenarios in chemical processes and industrial applications.