At \(0^{\circ} \mathrm{C}\) a \(1.0-\mathrm{L}\) flask contains \(5.0 \times 10^{-2} \mathrm{~mol} \mathrm{~N}_{2}, 1.5 \times 10^{2} \mathrm{mg}\) \(\mathrm{O}_{2}\), and \(5.0 \times 10^{21}\) molecules of \(\mathrm{NH}_{3} .\) What is the partial pressure of each gas, and what is the total pressure in the flask?

The Ideal Gas Law is a fundamental principle which relates the pressure, volume, temperature, and amount of a gas. Described by the equation \( PV = nRT \), where \( P \) represents the pressure, \( V \) is the volume, \( n \) is the number of moles of gas, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.

For students tackling partial pressure calculations, the Ideal Gas Law serves as a crucial tool. It helps predict how a gas will behave under different conditions and can be especially useful when dealing with gas mixtures in a confined space.

In the context of classroom exercises, solving for the pressure of an individual gas in a mixture, or its partial pressure, this law is rearranged to \( P = \frac{nRT}{V} \). When applied to each gas, this yields the partial pressures which can be added together to find the total pressure within a container. Since the pressure contribution of each gas is independent, this addition aligns with Dalton's Law of Partial Pressures.

The conversion of moles to mass and vice versa is critical in chemistry, especially when working with gases. To convert the mass of a substance to moles, you divide the mass by the molar mass (the mass of one mole) of the substance. The molar mass is numerically equal to the atomic or molecular weight expressed in grams.

Understanding this conversion allows students to translate the mass of a sample, often given in milligrams (\(mg\)) or grams (\(g\)), to moles (\(mol\)) which is integral for using the Ideal Gas Law effectively. For example, oxygen (\(O_2\)) has a molar mass of 32 g/mol, so if we have 0.15 g of \(O_2\), we divide by 32 g/mol to find the moles. This calculation is a key step delineated in many exercises that deal with gas mixtures and reactions.

Avogadro's number, approximately \(6.022 \times 10^{23}\) entities per mole, is a constant that represents the number of atoms, ions, or molecules in one mole of a substance. This figure is immensely useful when dealing with microscopic particles at a macroscopic level.

In gas law computations, particularly when we are given the quantity of gas in terms of individual molecules, Avogadro's number provides a bridge to convert this count to moles. As such, if a problem specifies a certain number of gas molecules, you can calculate the corresponding moles by dividing the number of molecules by Avogadro's number. This is frequently the first step before you can proceed to use the Ideal Gas Law for further calculations involving gases.