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 \(\mathrm{N}_{2} .^{\prime \prime}\) The pres- sure 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: H2S = 2.07 atm, CO2 = 1.38 atm, N2 = 42.55 atm.

Step 1: Understanding Partial Pressures

To find the partial pressure of each gas in a mixture, we use the formula \( P_i = y_i \cdot P_{total} \), where \( P_i \) is the partial pressure of the component, \( y_i \) is the mole fraction, and \( P_{total} \) is the total pressure.

Step 2: Calculate Total Mole Percentage of Nitrogen

Given that the total mole percentage of gases in the mixture is 100%, and the sum of the mole percentages of \( \mathrm{H}_2\mathrm{S} \) and \( \mathrm{CO}_2 \) are given as 4.5% and 3.0%, the remaining must be \( \mathrm{N}_2 \).Therefore, \( \% \mathrm{N}_2 = 100\% - 4.5\% - 3.0\% = 92.5\% \).

Step 3: Convert Mole Percentages to Mole Fractions

Mole percentages are converted to mole fractions by dividing by 100%.\( y_{\mathrm{H}_2\mathrm{S}} = \frac{4.5}{100} = 0.045 \) \( y_{\mathrm{CO}_2} = \frac{3.0}{100} = 0.030 \) \( y_{\mathrm{N}_2} = \frac{92.5}{100} = 0.925 \)

Step 4: Calculate Partial Pressure of H2S

Using the formula for partial pressure: \( P_{\mathrm{H}_2\mathrm{S}} = y_{\mathrm{H}_2\mathrm{S}} \times P_{total} \)\( P_{\mathrm{H}_2\mathrm{S}} = 0.045 \times 46 \text{ atm} = 2.07 \text{ atm} \).

Step 5: Calculate Partial Pressure of CO2

Using the formula for partial pressure:\( P_{\mathrm{CO}_2} = y_{\mathrm{CO}_2} \times P_{total} \)\( P_{\mathrm{CO}_2} = 0.030 \times 46 \text{ atm} = 1.38 \text{ atm} \).

Step 6: Calculate Partial Pressure of N2

Using the formula for partial pressure:\( P_{\mathrm{N}_2} = y_{\mathrm{N}_2} \times P_{total} \)\( P_{\mathrm{N}_2} = 0.925 \times 46 \text{ atm} = 42.55 \text{ atm} \).

Key Concepts

Dalton's Law of Partial Pressures

When dealing with gas mixtures, Dalton's Law of Partial Pressures is an essential concept. It explains how the total pressure of a gas mixture is the sum of the pressures that each gas in the mixture would exert if it occupied the entire volume by itself. The formula for the partial pressure of an individual gas component is:

• \( P_i = y_i \cdot P_{\text{total}} \)

Here:

• \( P_i \) is the partial pressure of a specific gas.
• \( y_i \) is the mole fraction of that gas in the mixture.
• \( P_{\text{total}} \) is the total pressure of the gas mixture.

This law allows us to determine how much pressure each gas contributes to the overall pressure in a system. It is widely applicable in chemistry and engineering to precisely gauge individual gas pressures in various processes.
Using Dalton’s Law helps to understand gas behaviors in mixed systems without needing to separate the gases physically.

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. It is defined as the ratio of the moles of one component to the total moles of all components in the mixture. Mole fraction has no units, as it is a simple ratio, making it convenient to use in various calculations.

• For a given gas, the mole fraction \( y_i \) is calculated as: \( y_i = \frac{\text{moles of gas } i}{\text{total moles of all gases}} \).

In practical terms:

• Mole percentage can be converted to mole fraction by dividing by 100.
• For instance, if the mole percentage of \( \text{H}_2\text{S} \) is 4.5%, its mole fraction is \( \frac{4.5}{100} = 0.045 \).

Using mole fractions simplifies calculations in gas law applications, allowing direct determination of partial pressures using Dalton's Law.

Gas Mixture Composition

Understanding the composition of a gas mixture involves knowing what gases are present and in what proportions. In the context of the cylinder problem, the labels indicate the composition in mole percentages. To work through problems involving gases, it's crucial to be able to convert these percentages into more useful formats like mole fraction.

• In the original exercise, the gas mixture is comprised of \( \text{H}_2\text{S} \), \( \text{CO}_2 \), and \( \text{N}_2 \).
• Each percentage represents how much each gas contributes to the whole in terms of moles.

For example:

• \( \text{H}_2\text{S} \) makes up 4.5 mole%, \( \text{CO}_2 \) makes up 3.0 mole%, and the rest is \( \text{N}_2 \) at 92.5 mole%.

By knowing the total pressure, you can employ Dalton's Law along with these mole fractions to find the partial pressures of each constituent gas. This understanding is foundational for fields that deal with gas mixtures, from atmospheric science to chemical engineering.