Problem 28 - Chemistry The Molecular Science

Question

A \(1.00-\mathrm{g}\) sample of water is allowed to vaporize completely inside a sealed \(10.0-\mathrm{L}\) container. Calculate the pressure of the water vapor at a temperature of \(150 .{ }^{\circ} \mathrm{C}\).

Step-by-Step Solution

Answer

The pressure of the water vapor is approximately 0.193 atm.

Step 1: Convert Units

First, convert the mass of the water sample from grams to moles. The molar mass of water is approximately 18.02 g/mol. Use the formula: \[ ext{moles of water} = \frac{1.00 \text{ g}}{18.02 \text{ g/mol}} \approx 0.0555 \text{ moles} \]

Step 2: Adjust Temperature

Convert the given temperature from Celsius to Kelvin, as the ideal gas law requires temperatures to be in Kelvin. \[ T(K) = 150 + 273.15 = 423.15 \text{ K} \]

Step 3: Apply Ideal Gas Law

Use the ideal gas law \( PV = nRT \) where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant (0.0821 L·atm/mol·K), and \( T \) is temperature in Kelvin. Rearrange to solve for \( P \): \[ P = \frac{nRT}{V} \]Substitute the values: \[ P = \frac{0.0555 \text{ moles} \times 0.0821 \text{ L·atm/mol·K} \times 423.15 \text{ K}}{10.0 \text{ L}} \approx 0.193 \text{ atm} \]

Key Concepts

Molar Mass of Water

Understanding the molar mass of water is essential when performing calculations involving water in chemical equations. The molar mass is essentially the mass of one mole of a substance. For water, the chemical formula is H₂O, meaning each molecule has 2 hydrogen atoms and 1 oxygen atom. Each hydrogen atom has an atomic mass of approximately 1.01 g/mol, and oxygen has a mass of about 16.00 g/mol. To find the molar mass of water, simply add these values together:

• 2 hydrogen atoms: 2 × 1.01 g/mol = 2.02 g/mol
• 1 oxygen atom: 16.00 g/mol

Putting these together, the molar mass of H₂O is 18.02 g/mol. This number is crucial because it allows us to convert from grams of water to moles, which is necessary when using the ideal gas law.

Temperature Conversion

Temperature conversion is another critical step when using the ideal gas law, as it requires temperature to be in Kelvin. Kelvin is the SI unit for temperature and is used because it ensures calculations are based on absolute temperature. Converting from Celsius to Kelvin involves a simple formula: \[ T(K) = T(°C) + 273.15 \]This conversion aligns Celsius temperature with Kelvin by accounting for the absolute zero point. In the given exercise, the temperature was initially 150 °C. Adding 273.15 yields 423.15 K. It’s crucial for students to convert temperature to Kelvin to avoid errors in calculations with gas laws.

Pressure Calculation

Pressure calculation is a fundamental part of applying the ideal gas law. The ideal gas law relation, \( PV = nRT \), can be rearranged to solve for pressure \( P \):\[ P = \frac{nRT}{V} \]Here, \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant (0.0821 L·atm/mol·K), and \( T \) is temperature in Kelvin.

• Insert the known values into the formula: \( n = 0.0555 \) moles, \( R = 0.0821 \), \( T = 423.15 \) K, and \( V = 10.0 \) L.
• Calculate: \( P = \frac{0.0555 \times 0.0821 \times 423.15}{10.0} \approx 0.193 \text{ atm} \)

This calculation shows how to find the pressure exerted by the water vapor under the given conditions. Understanding these steps helps ensure accurate results in any gas-related calculations.