A quantity of \(0.25 \mathrm{~g}\) of a substance when vaporized displaced \(50 \mathrm{~cm}^{3}\) of air at \(0^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm} .\) The gram molecular mass of the substance will be (a) \(50 \mathrm{~g}\) (b) \(100 \mathrm{~g}\) (c) \(112 \mathrm{~g}\) (d) \(127.5 \mathrm{~g}\)

When we're dealing with gases, one of the fundamental relationships is described by the Ideal Gas Law. This law is encapsulated in the simple formula PV = nRT, where P stands for pressure, V is volume, n is the number of moles of gas, R is the universal gas constant, and T is the temperature in Kelvin.

The ideal gas law allows us to relate the physical properties of a gas under given conditions, and it's particularly useful when we need to calculate the amount of gas in moles or understand how changes in conditions like temperature or pressure can affect a gas. In educational settings, it's commonly used for a variety of problem-solving exercises, such as the one provided.

Understanding molar mass is fundamental when dealing with chemical substances. The molar mass, often denoted by M_m, is the mass of one mole of a substance, usually expressed in grams per mole (g/mol).

To calculate the molar mass, we typically sum the atomic masses of all the atoms in a molecule. For example, the molar mass of water (H₂O) is approximately 18 g/mol because it has two hydrogen atoms and one oxygen atom. In exercises, the molar mass is crucial for converting between the mass of a substance and the number of moles. It’s often the key step in solving stoichiometry problems and is used in combination with the ideal gas law, as seen in our exercise.

Stoichiometry is a section of chemistry that involves calculating the quantities of reactants and products in a chemical reaction. It's based on the conservation of mass where the mass of the reactants equals the mass of the products. For stoichiometry calculations, the mole concept is essential since it allows chemists to work with amounts on a macroscopic level – grams, liters, etc. – rather than atomic scales.

In the context of our exercise, stoichiometry is applied to determine the gram molecular mass by relating the mass in grams to the number of moles using the Ideal Gas Law. This principle can be extended to complex chemical reactions, where the ratios of reactants and products, based on their mole ratios in a balanced equation, are crucial for accurate calculations.

The conversion between moles and grams is a common practice in chemistry and is crucial for both theoretical and practical applications. To convert moles to grams, you need to multiply the number of moles by the substance's molar mass (M_m). Conversely, to convert from grams to moles, divide the mass of the substance by the molar mass.

In our example, knowing the molar mass allowed us to convert the given mass of the substance (in grams) to the number of moles. This was essential to apply the Ideal Gas Law correctly. Whenever you're confronted with a problem involving mass and amount of a substance, this moles to grams conversion will be a vital step in finding your solution.