Question

What volume does a mixture of \(14.2 \mathrm{g}\) of He and \(21.6 \mathrm{g}\) of \(\mathrm{H}_{2}\) occupy at \(28^{\circ} \mathrm{C}\) and 0.985 atm?

Short answer

The volume of the mixture of Helium (He) and Hydrogen (H₂) gases is approximately 350.32 L at 28°C and 0.985 atm.

Step-by-step solution

Find the number of moles of each gas

To calculate the number of moles, divide the mass of each gas by its respective molar mass. The molar mass of Helium (He) is 4 g/mol, and the molar mass of Hydrogen (H₂) is 2 g/mol. For Helium: n(He) = mass(He) / molar mass(He) n(He) = 14.2 g / 4 g/mol n(He) = 3.55 mol For Hydrogen: n(H₂) = mass(H₂) / molar mass(H₂) n(H₂) = 21.6 g / 2 g/mol n(H₂) = 10.8 mol

Find the total number of moles in the mixture

To find the total number of moles, simply add the number of moles of each gas. n(total) = n(He) + n(H₂) n(total) = 3.55 mol + 10.8 mol n(total) = 14.35 mol

Convert temperature to Kelvin

The Ideal Gas Law uses temperature in Kelvin (K). To convert from Celsius (°C) to Kelvin (K), add 273.15. T = temperature in °C + 273.15 T = 28°C + 273.15 T = 301.15 K

Apply the Ideal Gas Law

The Ideal Gas Law is given by the formula: PV = nRT. We have the values for the number of moles (n), temperature (T), and pressure (P). The only missing variable is the volume (V). Also, note that the Gas Constant R = 0.0821 L atm/mol K. 0.985 atm * V = 14.35 mol * 0.0821 L atm/mol K * 301.15 K Now, solve for V. V = (14.35 mol * 0.0821 L atm/mol K * 301.15 K) / 0.985 atm V ≈ 350.32 L

Present the final answer

The volume of the mixture of Helium (He) and Hydrogen (H₂) gases is approximately 350.32 L at 28°C and 0.985 atm.

Key concepts

Moles Calculation

When dealing with gases, calculating the number of moles is an essential first step. The number of moles, symbolized by the letter "n," tells us how many atoms, ions, or molecules are present in a given substance. This calculation is crucial for using the Ideal Gas Law to determine other properties of the gas.

To calculate the number of moles, you need the mass of the substance and its molar mass. The formula used is:

• Number of moles \( n = \frac{\text{mass of the substance}}{\text{molar mass}} \)

For example, in the given problem, the number of moles of Helium (He) is calculated from its mass (14.2 g) and its molar mass (4 g/mol) as follows:

• \( n(\text{He}) = \frac{14.2 \, \text{g}}{4 \, \text{g/mol}} = 3.55 \, \text{mol} \)

Likewise, the number of moles of Hydrogen (H\(_2\)) is determined using its mass (21.6 g) and its molar mass (2 g/mol):

• \( n(\text{H}_2) = \frac{21.6 \, \text{g}}{2 \, \text{g/mol}} = 10.8 \, \text{mol} \)

By summing the moles of He and H\(_2\), we find the total number of moles in this gas mixture is 14.35 mol. This total is crucial for the Ideal Gas Law calculations.

Temperature Conversion to Kelvin

In many scientific calculations, including the Ideal Gas Law, temperature must be in Kelvin. Kelvin (K) is the SI unit of temperature and provides a true measure of thermal energy.

Unlike Celsius, Kelvin starts at absolute zero, the point at which all molecular motion ceases. Because of its absolute nature, Kelvin does not have negative values, which simplifies many calculations.

To convert temperature from Celsius (°C) to Kelvin (K), you simply add 273.15:

• \( T(\text{K}) = T(°\text{C}) + 273.15 \)

So, for this problem where the temperature is 28°C:

• \( T = 28°C + 273.15 = 301.15 \, \text{K} \)

This temperature conversion is vital as it ensures that calculations involving the Ideal Gas Law remain accurate and consistent.

Gas Constant R

In the Ideal Gas Law, the gas constant "R" is a crucial factor that helps relate pressure, volume, and temperature to the number of moles of a gas. It acts as a bridge, ensuring the units of each component in the equation \( PV = nRT \) are compatible.

The commonly used value for R is 0.0821 L atm/mol K. This version of R is specifically tailored for use when pressure is measured in atmospheres (atm), volume in liters (L), and temperature in Kelvin (K). It's important to always match the units of your measurements to those required by R to ensure the integrity of your calculations.

In the provided problem, R facilitates the calculation of gas volume under certain conditions:

• Given that the number of moles (n) is 14.35 mol, the pressure (P) is 0.985 atm, and the temperature (T) is 301.15 K, you would solve for the volume (V) like this: \( V = \frac{14.35 \times 0.0821 \times 301.15}{0.985} \)

This calculation demonstrates the utility of the gas constant and allows us to solve for missing variables in gas-related problems efficiently.