Question

Which of the following gases has a density of \(2.14 \mathrm{~g} / \mathrm{L}\) at \(\mathrm{STP:} \mathrm{H}_{2}, \mathrm{O}_{2}, \mathrm{O}_{3} ?\)

Short answer

\(\mathrm{O}_3\) has a density of 2.14 g/L at STP.

Step-by-step solution

Understand the Problem

We need to determine which gas among \( \mathrm{H}_2, \mathrm{O}_2, \mathrm{O}_3 \) has a density of \( 2.14 \, \mathrm{g/L} \) at standard temperature and pressure (STP). At STP, the molar volume of any gas is \( 22.4 \, \mathrm{L/mol} \).

Calculate Molar Mass of Each Gas

Calculate the molar mass of each gas: - \( \mathrm{H}_2 \): Molar Mass = \(2 \times 1.01 = 2.02 \, \mathrm{g/mol} \).- \( \mathrm{O}_2 \): Molar Mass = \(2 \times 16.00 = 32.00 \, \mathrm{g/mol} \).- \( \mathrm{O}_3 \): Molar Mass = \(3 \times 16.00 = 48.00 \, \mathrm{g/mol} \).

Apply Density Formula for Each Gas

For each gas, use the formula \( \text{Density} = \frac{\text{Molar Mass}}{\text{Molar Volume}} \) to find the density at STP:- \( \mathrm{H}_2 \): \[ \text{Density} = \frac{2.02 \, \mathrm{g/mol}}{22.4 \, \mathrm{L/mol}} = 0.090 \, \mathrm{g/L} \]- \( \mathrm{O}_2 \): \[ \text{Density} = \frac{32.00 \, \mathrm{g/mol}}{22.4 \, \mathrm{L/mol}} = 1.43 \, \mathrm{g/L} \]- \( \mathrm{O}_3 \): \[ \text{Density} = \frac{48.00 \, \mathrm{g/mol}}{22.4 \, \mathrm{L/mol}} = 2.14 \, \mathrm{g/L} \]

Compare Calculated Density with Given Density

Compare the calculated densities of each gas with the given density of \(2.14 \, \mathrm{g/L} \). \( \mathrm{O}_3 \) is the only gas with the correct density.

Key concepts

Molar Mass Calculation

The molar mass of a gas is an important value in understanding its density at standard conditions. It is determined by the mass of one mole of molecules of the gas. To calculate the molar mass, you sum up the atomic masses of all atoms present in one molecule of the gas.
For instance:

• The molar mass of hydrogen (\( \mathrm{H}_2 \)) is calculated as: \( 2 \times 1.01 = 2.02 \, \mathrm{g/mol} \). This is because each hydrogen atom has an atomic mass of approximately 1.01 g/mol.
• For oxygen (\( \mathrm{O}_2 \)), the molar mass is: \( 2 \times 16.00 = 32.00 \, \mathrm{g/mol} \).
• And for ozone (\( \mathrm{O}_3 \)), the molar mass is:\( 3 \times 16.00 = 48.00 \, \mathrm{g/mol} \).

Knowing how to calculate the molar mass is essential for determining other properties of the gas, such as its density.

Standard Temperature and Pressure

Standard temperature and pressure (STP) is a set of conditions used to make comparisons between different gases simpler. STP is defined as a temperature of 273.15 K (0°C) and a pressure of 1 atm (101.325 kPa). At STP, gases behave in a predictable way, making calculations of gas properties easier.

The molar volume of an ideal gas at STP is always \( 22.4 \, \mathrm{L/mol} \). This is used because it is often cumbersome to deal with specific experimental conditions for every gas. The molar volume is crucial when working with gases at STP as it relates directly to density calculations, chemical reactions, and conversions between moles and volume.

Density Formula

Density is a crucial concept when dealing with gases and is represented as the mass of the gas per unit volume. The formula to calculate the density (\( \rho \)) of a gas is given by:\[ \text{Density} = \frac{\text{Molar Mass}}{\text{Molar Volume}} \]\This formula helps us find out how many grams of the gas occupy one liter at specified conditions like STP.

For example, if you want to find out if a gas has a density of \( 2.14 \, \mathrm{g/L} \) at STP, you can calculate its expected density using the above formula.

• For \( \mathrm{H}_2 \), the density would be: \( \frac{2.02 \, \mathrm{g/mol}}{22.4 \, \mathrm{L/mol}} = 0.090 \, \mathrm{g/L} \).
• For \( \mathrm{O}_2 \), the density calculates as: \( \frac{32.00 \, \mathrm{g/mol}}{22.4 \, \mathrm{L/mol}} = 1.43 \, \mathrm{g/L} \).
• Meanwhile, for \( \mathrm{O}_3 \), the density is: \( \frac{48.00 \, \mathrm{g/mol}}{22.4 \, \mathrm{L/mol}} = 2.14 \, \mathrm{g/L} \).

Thus, based on this formula, one can determine which gas matches the given density values.

Molar Volume

Molar volume is a measure of the volume occupied by one mole of a gas. At standard temperature and pressure (STP), all ideal gases have the same molar volume, which is \( 22.4 \, \mathrm{L/mol} \).

This consistent value allows us to easily calculate the density of gases and predict their behavior under these standard conditions. Molar volume makes it possible to simplify the complex movements of countless gas molecules into a manageable number suitable for calculations.

• This means that if you have one mole of \( \mathrm{O}_3 \), it will occupy \( 22.4 \, \mathrm{L} \), just like \( \mathrm{H}_2 \) or \( \mathrm{O}_2 \) would, despite differences in molecular complexity and weight.
• Using the molar volume at STP simplifies many calculations involving gases, such as predicting the outcome of reactions or determining gas density from molar mass.

Understanding and using molar volume correctly enables accurate predictions and calculations in gas-related problems.