Question

(a) What are the mole fractions of each component in a mixture of \(15.08 \mathrm{~g}\) of \(\mathrm{O}_{2}, 8.17 \mathrm{~g}\) of \(\mathrm{N}_{2}\), and \(2.64 \mathrm{~g}\) of \(\mathrm{H}_{2} ?\) (b) What is the partial pressure in atm of each component of this mixture if it is held in a 15.50-L vessel at \(15^{\circ} \mathrm{C}\) ?

Short answer

(a) The mole fractions of O₂, N₂, and H₂ in the mixture are approximately 0.226, 0.140, and 0.634, respectively. (b) The partial pressures of O₂, N₂, and H₂ in the 15.50-L vessel at 15ºC are approximately 0.56 atm, 0.35 atm, and 1.56 atm, respectively.

Step-by-step solution

Calculate the moles of each component

To calculate the moles of each component, we need to divide the mass given for each substance by its corresponding molar mass. The molar masses are: O₂ = 32 g/mol, N₂ = 28 g/mol, and H₂ = 2 g/mol. For O₂, moles_O₂ = \( \frac{15.08 g}{32 g/mol} = 0.47125 \; mol\) For N₂, moles_N₂ = \( \frac{8.17 g}{28 g/mol} = 0.29196 \; mol\) For H₂, moles_H₂ = \( \frac{2.64 g}{2 g/mol} = 1.32 \; mol\)

Calculate total moles in the mixture

To calculate the total moles in the mixture, simply add the moles of each component (n_total = moles_O₂ + moles_N₂ + moles_H₂): n_total = 0.47125 mol + 0.29196 mol + 1.32 mol = 2.08321 mol

Calculate the mole fraction of each component

To calculate the mole fraction of each component, divide the moles of the component by the total moles in the mixture: mole_fraction_O₂ = \( \frac{0.47125 mol}{2.08321 mol} = 0.22617 \) mole_fraction_N₂ = \( \frac{0.29196 mol}{2.08321 mol} = 0.14010 \) mole_fraction_H₂ = \( \frac{1.32 mol}{2.08321 mol} = 0.63373 \)

Calculate the total pressure of the mixture

For part (b), we will use the ideal gas law PV=nRT to calculate the total pressure of the mixture. We are given the temperature (15ºC) and the volume (15.50 L). First, convert temperature to Kelvin: T(K) = 15ºC + 273.15 = 288.15 K. The ideal gas constant, R, equals 0.0821 L·atm/mol·K. Now, we can plug the numbers into the equation: P_total * 15.50 L = 2.08321 mol * 0.0821 L·atm/mol·K * 288.15 K Now, solve for P_total: P_total = \( \frac{2.08321 mol * 0.0821 L·atm/mol·K * 288.15 K}{15.50 L} = 2.46883 \; atm\)

Calculate the partial pressure of each component

To calculate the partial pressure of each component, multiply the mole fraction of the component by the total pressure: Partial_pressure_O₂ = 0.22617 * 2.46883 atm = 0.55832 atm Partial_pressure_N₂ = 0.14010 * 2.46883 atm = 0.34549 atm Partial_pressure_H₂ = 0.63373 * 2.46883 atm = 1.56467 atm So, the partial pressures of O₂, N₂, and H₂ are approximately 0.56 atm, 0.35 atm, and 1.56 atm, respectively.

Key concepts

Partial Pressure

When studying gas mixtures, it's crucial to understand the concept of partial pressure. This refers to the pressure that each gas in a mixture would exert if it were alone in the container at the same temperature. Since gases in a mixture behave independently according to Dalton's Law of Partial Pressures, the total pressure exerted by the gas mixture is the sum of the partial pressures of each gas within it.

For a practical understanding, imagine a room filled with different scents. Each scent represents a gas in the mixture, and the intensity with which you can smell each one depends on its concentration—the more molecules of a scent, the stronger it is. Similarly, the more molecules of a specific gas, the higher the partial pressure. To calculate it, as shown in the exercise solution, you multiply the mole fraction of the gas by the total pressure of the mixture. Remember, the mole fraction is simply the ratio of the number of moles of a particular gas to the total number of moles in the mixture.

Ideal Gas Law

The Ideal Gas Law, represented by the equation PV=nRT, is a cornerstone principle in chemistry that relates the pressure (P), volume (V), and temperature (T) of an ideal gas to the amount of substance in moles (n) and a constant (R). The R is known as the ideal gas constant and varies depending on the units of pressure, volume, and temperature. In this context, for instance, R is given as 0.0821 L·atm/mol·K.

The beauty of this equation is its ability to adapt to various gas-related problems. Once you know any three variables, you can find the fourth. For the given exercise, we used the Ideal Gas Law to find the total pressure exerted by the gas mixture. To ensure accuracy in your calculations, always convert temperature to Kelvin and check your units across the equation.

Molar Mass

Molar mass is like the weight on a baker's recipe—it tells you how much one mole of a substance weighs. It's a vital concept for converting between grams and moles, which allows chemists to use the mole as a counting unit for atoms and molecules, much like using the dozen to count eggs. Every element has a unique molar mass, listed on the periodic table, typically in units of grams per mole (g/mol).

In our exercise, the molar mass enables us to start with the mass of each gas (in grams) and convert it to its equivalent in moles. It provides the groundwork for calculating mole fractions and, subsequently, partial pressures in gas mixtures. Understanding molar mass is not only crucial for such calculations but also for a myriad of other chemical quantifications, from stoichiometry to solution concentrations.