(a) How many moles of hydrogen will be liberated from \(200.0 \mathrm{~mL}\) of \(3.00 \mathrm{M} \mathrm{HCl}\) reacting with an excess of magnesium? The equation is $$ \mathrm{Mg}(s)+2 \mathrm{HCl}(a q) \longrightarrow \mathrm{MgCl}_{2}(a q)+\mathrm{H}_{2}(g) $$ (b) How many liters of hydrogen gas \(\left(\mathrm{H}_{2}\right)\) measured at \(27^{\circ} \mathrm{C}\) and 720 torr will be obtained? (Hint: Use the ideal gas law.)

The mole concept is a fundamental principle in chemistry that serves as a bridge between the microscopic world of atoms and molecules and the macroscopic world of grams and liters. It enables chemicians to count entities at the atomic scale by using Avogadro's number, which is approximately \(6.022 \times 10^{23}\) entities per mole. In practice, the mole concept allows us to calculate the number of particles in a given substance when we know its mass, or conversely, to deduce the mass when the number of moles is known.

For example, when dealing with the reaction of hydrochloric acid (HCl) and magnesium (Mg), as in the textbook exercise, understanding moles is essential. The balanced chemical equation shows the mole-to-mole ratio of reactants and products, allowing us to predict the amounts of substances involved in the reaction.

Molarity, a measure of concentration, is defined as the number of moles of a solute divided by the volume of the solution in liters. It is commonly expressed in moles per liter (M). This concept is pivotal when mixing chemical solutions or determining reactant quantities. A typical molarity calculation involves using the formula M = moles of solute / liters of solution.

To illustrate, in given exercise (a), the molarity of HCl is used to find out how much hydrogen gas will be produced. By knowing the molarity and the volume of the hydrochloric acid, we can calculate the moles of HCl present, which then informs us of the moles of hydrogen gas that can be generated.

Chemical reaction stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. It involves the use of balanced chemical equations to determine the proportions of reactants needed or products formed. The coefficients in a balanced equation provide the ratio of moles of each substance involved.

In this context, when magnesium reacts with hydrochloric acid, stoichiometry tells us that one mole of magnesium will react with two moles of HCl to produce magnesium chloride (MgCl2) and hydrogen gas (H2). With stoichiometry, we can predict that for every two moles of HCl used, one mole of H2 gas will be produced. This ratio is essential to solve part (a) of our exercise.

The ideal gas law is a cornerstone of gas laws, expressed as PV = nRT, where P represents the pressure of the gas, V is its volume, n is the number of moles, R is the universal gas constant, and T is the temperature in Kelvin. This law describes the behavior of ideal gases by correlating these four variables.

In real-life applications, such as exercise (b), we use the ideal gas law to find the volume of a gas at a specific temperature and pressure once the amount (in moles) is known. The universal gas constant (R) has a value of 0.0821 L*atm/mol*K. Converting the given temperature to Kelvin and the pressure to atmospheres enables us to use this law to calculate the volume of hydrogen gas produced in the reaction.

Gas volume calculation involves finding the volume occupied by a gas under certain conditions of temperature and pressure, and it can be performed using the ideal gas law. When conditions are not at standard temperature and pressure (STP), conversion to the appropriate units is necessary to apply the ideal gas law accurately.

In the exercise, after determining the number of moles of hydrogen gas and converting the given pressure and temperature to their correct units (atmospheres and Kelvin, respectively), we can calculate the volume of the gas. This calculation is particularly important in chemical engineering and environmental studies, where the volume of gases released or needed in processes must be controlled and measured.