(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 \(15^{\circ} \mathrm{C}\) ?

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.

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 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.