A 0.12-mol sample of nitrogen gas occupies a volume of \(2.55 \mathrm{~L}\). What is the volume of \(0.32 \mathrm{~mol}\) of nitrogen gas under the same conditions?

Understanding the molar volume of a gas is crucial when studying chemistry because it relates the volume occupied by a gas to the amount of substance present, typically measured in moles. At standard temperature and pressure (STP), which is defined as 0 degrees Celsius and 1 atmosphere of pressure, one mole of any ideal gas occupies 22.4 liters. This is known as the molar volume of a gas at STP. In practical scenarios, conditions may vary, and thus the volume occupied by a mole of gas changes accordingly.

When applying Avogadro's law, which states that the volume of a gas is directly proportional to the number of moles of gas, given that temperature and pressure are held constant, we can calculate changes in volume during chemical reactions or when different amounts of substances are involved. For instance, the problem given establishes that 0.12 moles of nitrogen gas occupy 2.55 liters. Knowing the molar volume allows for the calculation of expected volumes at other quantities of the gas while maintaining the same conditions.

Direct proportionality is a straightforward but powerful concept in chemistry. It indicates a relationship between two variables where if one increases, the other increases by the same factor. This is expressed mathematically as 'y is directly proportional to x,' or y ∝ x. In a formula, it would appear as y = kx, where k is the constant of proportionality.

In the context of Avogadro's law, volume (V) is directly proportional to the number of moles (n), and this relationship is fundamental to solving problems involving gas volumes and quantities. In our exercise, we see direct proportionality in action: as the number of moles of nitrogen gas increases from 0.12 to 0.32, the volume also increases, maintaining a consistent ratio. This proportionality simplifies the process of determining unknown volumes or moles when the other variable is known, and is a cornerstone concept in chemical calculations.

Stoichiometry is a branch of chemistry that deals with the quantitative relationships between the reactants and products in a chemical reaction. It is based on the conservation of mass where the mass of the reactants equals the mass of the products. Stoichiometry uses the balanced chemical equation to determine the amounts of substances needed or produced in a reaction.

To apply stoichiometry to gases, we often use Avogadro's law to relate the volumes of gases at the same temperature and pressure to the coefficients in the balanced equation. For instance, in our example of nitrogen gas, if we were to react nitrogen with another substance, stoichiometry would allow us to predict the volume of nitrogen required or produced based on the stoichiometric coefficients from the balanced equation and the concept of molar volume. This integration of concepts demonstrates the interconnectedness of stoichiometry, molar volume, and direct proportionality in solving complex chemical problems.