Problem 74 - Chemistry The Central Science In Si Units

Question

A sample of \(3.00 \mathrm{~g}\) of \(\mathrm{SO}_{2}(g)\) originally in a \(5.00-\mathrm{L}\) vessel at \(21^{\circ} \mathrm{C}\) is transferred to a \(10.0-\mathrm{L}\) vessel at \(26^{\circ} \mathrm{C}\). A sample of \(2.35 \mathrm{~g}\) of \(\mathrm{N}_{2}(g)\) originally in a \(2.50-\mathrm{L}\) vessel at \(20^{\circ} \mathrm{C}\) is transferred to this same \(10.0-\mathrm{L}\) vessel. (a) What is the partial pressure of \(\mathrm{SO}_{2}(g)\) in the larger container? (b) What is the partial pressure of \(\mathrm{N}_{2}(g)\) in this vessel? (c) What is the total pressure in the vessel?

Step-by-Step Solution

Answer

The partial pressure of SO2 is 0.115 atm, the partial pressure of N2 is 0.206 atm, and the total pressure is 0.321 atm.

Step 1: Calculate Initial Moles of SO2

First, calculate the number of moles of SO2 using its molar mass. The molar mass of SO2 is approximately 64.07 g/mol. Use the formula: \( n = \frac{m}{M} \) where \( m = 3.00 \text{ g} \) and \( M = 64.07 \text{ g/mol} \). This gives: \( n = \frac{3.00}{64.07} = 0.0468 \text{ mol} \).

Step 2: Apply Ideal Gas Law for SO2 in New Conditions

Next, use the ideal gas law in the new conditions to find the partial pressure \( P \) of SO2 in the 10.0-L vessel at 26°C. Convert the temperature to Kelvin: \( T = 26 + 273.15 = 299.15 \text{ K} \). The ideal gas equation is \( PV = nRT \). Solve for \( P \): \( P = \frac{nRT}{V} = \frac{0.0468 \times 0.0821 \times 299.15}{10.0} = 0.115 \text{ atm} \).

Step 3: Calculate Initial Moles of N2

Calculate the number of moles of N2 using its molar mass. The molar mass of N2 is approximately 28.02 g/mol. Use the formula: \( n = \frac{m}{M} \) where \( m = 2.35 \text{ g} \) and \( M = 28.02 \text{ g/mol} \). This gives: \( n = \frac{2.35}{28.02} = 0.0839 \text{ mol} \).

Step 4: Apply Ideal Gas Law for N2 in New Conditions

Use the ideal gas law for N2 in the new vessel conditions. The temperature 26°C is converted to Kelvin: \( T = 299.15 \text{ K} \). Use the ideal gas law: \( P = \frac{nRT}{V} = \frac{0.0839 \times 0.0821 \times 299.15}{10.0} = 0.206 \text{ atm} \).

Step 5: Calculate Total Pressure in the Vessel

The total pressure in the vessel is the sum of the partial pressures of SO2 and N2. Total pressure \( P_{total} = P_{SO2} + P_{N2} = 0.115 + 0.206 = 0.321 \text{ atm} \).

Key Concepts

Partial Pressure

Partial pressure is an essential concept when dealing with gas mixtures. It represents the pressure exerted by an individual gas within a mixture if it alone occupied the entire volume of the container, under the same temperature conditions. This concept is derived from Dalton's Law of Partial Pressures, which states that the total pressure of a gas mixture is the sum of the partial pressures of each individual gas. Let's consider an example with gases SO₂ and N₂ in a 10.0-L vessel. To find the partial pressure of SO₂, we use the ideal gas law formula: \[ P = \frac{nRT}{V} \]Where:

• \( n \) is the number of moles of the gas,
• \( R \) is the ideal gas constant (0.0821 atm•L/mol•K),
• \( T \) is the temperature in Kelvin, and
• \( V \) is the volume of the container.

By plugging in the values for SO₂ at a new temperature of 299.15 K (converted from 26°C), you can compute its partial pressure as described in the instructions above. This process can be repeated for N₂ to find its partial pressure. Understanding partial pressure helps in predicting how gases behave in mixtures, which is crucial in many scientific fields.

Moles Calculation

Calculating moles is key for determining how gases behave under different conditions. The mole is a standard scientific unit used to measure large quantities of very small entities, such as atoms or molecules. To find the number of moles, you can use the formula:\[ n = \frac{m}{M} \]Where:

• \( n \) is the number of moles,
• \( m \) is the mass of the substance (in grams), and
• \( M \) is the molar mass of the substance (grams per mole).

For instance, given 3.00 g of SO₂, and knowing that the molar mass of SO₂ is approximately 64.07 g/mol,\[ n = \frac{3.00}{64.07} = 0.0468 \text{ mol} \]This calculation allows you to determine the number of molecules present, enabling further calculations, such as determining the partial pressure using the ideal gas law. Moles calculation is fundamental in bridging the gap between macroscopic measurements and microscopic quantities in chemistry.

Temperature Conversion

Temperature conversion is a critical initial step in gas law calculations. Gases behave differently at different temperatures, and the ideal gas law formula \( PV = nRT \) requires that temperature is in Kelvin. This is because Kelvin is an absolute temperature scale based on the molecular motion, where 0 K is absolute zero, the point at which molecular motion ceases entirely.To convert Celsius to Kelvin, use the formula:\[ T(K) = T(°C) + 273.15 \]For example, an initial temperature of 26°C must be converted to Kelvin:\[ T = 26 + 273.15 = 299.15 \text{ K} \]This converted temperature is then used in subsequent calculations involving the ideal gas law. Properly converting temperature ensures accurate calculations and predictions of gas behavior. Missteps in conversion can lead to significant errors, as the precision of gas law calculations hinges on accurate temperature values.