In molar calculations involving gases, the ideal gas equation is a critical tool. The equation is PV = nRT, where P is 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. This equation allows the calculation of one of these variables if the others are known. For example, if the pressure, volume, and temperature of a certain amount of gas are known, the number of moles of the gas can be calculated. This is particularly useful in laboratory situations where the conditions of a gas (pressure, volume, temperature) are controlled or measured. The ideal gas equation also helps in converting between different units of gas volumes, given that conditions of temperature and pressure are stated. It's important to note that the ideal gas law assumes that the gas behaves ideally, meaning the gas particles do not interact with each other, and the volume of the gas particles themselves is negligible compared to the volume of the container. While real gases do not perfectly adhere to these assumptions, the ideal gas law often provides a good approximation for gas behavior under many conditions.

The concept of moles is fundamental in determining the stoichiometry of chemical reactions. Stoichiometry is the study of the quantitative relationships or ratios between reactants and products in a chemical reaction. In a balanced chemical equation, the coefficients indicate the ratio of moles of each reactant and product. For instance, in the reaction 2H₂ + O₂ → 2H₂O, the coefficients 2, 1, and 2 tell us that 2 moles of hydrogen react with 1 mole of oxygen to produce 2 moles of water. This molar ratio is essential for calculating how much of each reactant is needed to produce a desired amount of product, or how much product can be expected from a given amount of reactants. It enables chemists to predict the outcomes of chemical reactions in quantitative terms, ensuring that reactants are used efficiently, and helps in determining the limiting reactant - the reactant that will be entirely consumed first and thus limits the amount of product formed.

The mole concept can indeed be applied to both elements and compounds, serving as a universal method for expressing quantities in chemistry. For elements, a mole corresponds to Avogadro's number of atoms of that element. The atomic mass of an element in grams per mole (g/mol) provides a direct link between the microscopic scale (individual atoms) and the macroscopic scale (grams). For example, 12 g of carbon-12 represents one mole of carbon atoms.

For compounds, a mole relates to Avogadro's number of molecules of that compound. The molar mass of a compound, which is the sum of the atomic masses of all the atoms in a molecule of the compound, is used to convert between grams and moles. For instance, the molar mass of water (H₂O) is 18 g/mol (2×1 for hydrogen + 16 for oxygen), meaning that 18 g of water is one mole of water molecules. This universal applicability of the mole concept allows chemists to perform calculations and make predictions about reactions and properties of substances, whether they are dealing with pure elements or complex compounds.

Molar calculations are integral in determining the concentration of a solution. Concentration in chemistry refers to the amount of a substance in a given volume of solution. The most common unit of concentration is molarity, which is defined as moles of solute per litre of solution (mol/dm³). To calculate the molarity of a solution, the number of moles of the solute must be determined, typically through stoichiometric calculations. This can involve using the molar mass of the solute to convert from mass to moles. Once the number of moles is known, the molarity is calculated by dividing the moles of solute by the volume of the solution in litres. For example, if you dissolve 58.5 g of NaCl (molar mass = 58.5 g/mol) in water to make 0.5 dm³ of solution, first find the moles of NaCl (58.5 g ÷ 58.5 g/mol = 1 mol), and then calculate the molarity (1 mol ÷ 0.5 dm³ = 2 M). This process is vital in laboratory settings for preparing solutions with precise concentrations for experiments.

The relationship between the number of moles of a gas and its temperature at constant pressure is described by Gay-Lussac's Law. This law states that the volume of a gas is directly proportional to its temperature (measured in Kelvin) when the pressure and the number of moles of the gas remain constant. Therefore, at constant pressure, an increase in temperature will cause an increase in the volume of the gas. Since the number of moles of a gas is directly proportional to its volume (Avogadro's Law), an increase in temperature, assuming no gas escapes, does not affect the number of moles. The moles of gas remain constant as temperature changes, provided the gas is contained in a flexible container that allows volume change and no gas is added or removed. It's essential to remember that these relationships are ideal and assume ideal gas behavior, which may not accurately represent real gases under all conditions.