Question

Calculate the total pressure of a mixture of \(1.50 \mathrm{~g} \mathrm{H}_{2}\) and \(5.00 \mathrm{~g} \mathrm{~N}_{2}\) in a sealed \(5.0-\mathrm{L}\) vessel at \(25^{\circ} \mathrm{C}\).

Short answer

The total pressure is 4.518 atm.

Step-by-step solution

Convert Mass to Moles

First, we need to convert the mass of each gas to moles using their molar masses. The molar mass of hydrogen (\(\mathrm{H_2}\)) is approximately \(2.02\, \mathrm{g/mol}\), and for nitrogen (\(\mathrm{N_2}\)) it is \(28.02\, \mathrm{g/mol}\).\[ n_{\mathrm{H_2}} = \frac{1.50\, \mathrm{g}}{2.02\, \mathrm{g/mol}} = 0.7426\, \mathrm{mol} \]\[ n_{\mathrm{N_2}} = \frac{5.00\, \mathrm{g}}{28.02\, \mathrm{g/mol}} = 0.1785\, \mathrm{mol} \]

Calculate the Total Moles of Gas

Add the moles of hydrogen and nitrogen to find the total number of moles of gas in the mixture.\[ n_{\text{total}} = n_{\mathrm{H_2}} + n_{\mathrm{N_2}} = 0.7426 + 0.1785 = 0.9211\, \mathrm{mol} \]

Use Ideal Gas Law to Find Total Pressure

Apply the ideal gas law to calculate the total pressure of the gas mixture. The ideal gas law is \(PV = nRT\), where \(P\) is the pressure, \(V\) is the volume (5.0 L), \(n\) is the total moles (0.9211 mol), \(R\) is the gas constant (0.0821 L·atm/mol·K), and \(T\) is the temperature in Kelvin (25°C + 273 = 298 K).\[ P = \frac{nRT}{V} = \frac{0.9211 \times 0.0821 \times 298}{5.0} = 4.518\, \mathrm{atm} \]

Key concepts

Gas Mixtures

When we talk about gas mixtures, we're referring to a combination of different gases sharing the same container. In our exercise, we have a mixture composed of hydrogen ( H_2 ) and nitrogen ( N_2 ). Each gas in the mix behaves independently, according to its own properties, but collectively, they obey the ideal gas laws when mixed. This independence means each gas contributes to the total pressure of the system. Gas mixtures are common in various scientific and industrial processes. They allow us to exploit the properties of different gases simultaneously.

• In the case of air, it is essentially a mixture of gases like nitrogen, oxygen, and minor components like argon and carbon dioxide.
• While working with gas mixtures, it's critical to assume the gases do not chemically react with each other, which simplifies calculations involving pressure and volume.

Moles Calculation

The calculation of moles is an essential step in working with gases, as it tells us how much substance we are dealing with. The mole is a standard SI unit used to measure the amount of a substance, and it is based on the number of atoms in exactly 12 grams of carbon-12.To calculate moles, we use the concept of molar mass, which is the mass of one mole of a given substance. For example:

• The molar mass of H_2 is approximately 2.02 g/mol.
• For N_2, the molar mass is about 28.02 g/mol.

To convert grams to moles, you divide the mass of the substance by its molar mass. For instance:- For hydrogen: \( n_{ ext{H}_2} = \frac{1.50 \, \text{g}}{2.02 \, \text{g/mol}} = 0.7426 \, \text{mol} \)- For nitrogen: \( n_{ ext{N}_2} = \frac{5.00 \, \text{g}}{28.02 \, \text{g/mol}} = 0.1785 \, \text{mol} \)After finding the moles of each individual gas in the mixture, sum them to find the total moles of gas present. This total is crucial for further calculations, such as pressure using the ideal gas law.

Molar Mass

Molar mass is a fundamental concept in chemistry that represents the mass of one mole of a substance. It's typically expressed in grams per mole (g/mol) and is numerically equivalent to the average atomic or molecular mass of the substance. Understanding molar mass is crucial because it directly links the mass of a substance to the amount of substance or moles you have.

• For elements, the molar mass is simply the atomic mass found on the periodic table. For instance, hydrogen has an atomic mass of approximately 1.01 amu, leading to a molar mass of 2.02 g/mol for H_2 .
• For compounds, you calculate the molar mass by summing the molar masses of the individual elements that make up the compound. For example, in N_2 , each nitrogen atom has an approximate atomic mass of 14.01 amu, resulting in a molar mass of 28.02 g/mol for diatomic nitrogen.

Molar mass serves as the conversion factor between a substance's mass and its moles. Using molar mass allows for accurate and consistent stoichiometric calculations in chemical reactions and gas law-related problems.