One of the half-reactions for the electrolysis of water is $$2 \mathrm{H}^{+}(a q)+2 e^{-} \longrightarrow \mathrm{H}_{2}(g)$$ If \(0.845 \mathrm{~L}\) of \(\mathrm{H}_{2}\) is collected at \(25^{\circ} \mathrm{C}\) and \(782 \mathrm{mmHg}\), how many moles of electrons had to pass through the solution?

When studying the properties of gases, the ideal gas law serves as a fundamental equation to understand how factors like pressure, volume, temperature, and the amount of gas are related. This law is usually expressed as the equation \( PV = nRT \), where \( P \) stands for pressure, \( V \) is volume, \( n \) represents the number of moles of gas, \( R \) is the ideal gas constant, and \( T \) denotes temperature in Kelvin.

In the context of electrolysis of water, the ideal gas law allows us to calculate the number of moles of hydrogen gas formed by using the pressure, volume, and temperature from the experiment. To use the ideal gas law accurately, it's important to convert the units appropriately, like pressure from mmHg to atm by dividing by 760. Also, since temperature needs to be in Kelvin for gas law calculations, we add 273.15 to the Celsius temperature.

For electrical reactions such as electrolysis, understanding the flow of electrons is key. Moles of electrons represent the quantity of electricity involved in such reactions. One mole of electrons is also known as a Faraday, and it is equivalent to the charge of approximately \(6.022 \times 10^{23}\) electrons (Avogadro’s number).

Considering the half-reaction from the electrolysis of water, we see that \(2\) moles of electrons reduce \(2\) protons to form \(1\) mole of hydrogen gas (\(\mathrm{H}_2\)). This stoichiometric relationship between the moles of electrons and the moles of hydrogen produced is vital for determining the amount of electrical charge needed to produce a certain amount of hydrogen gas.

Gas law calculations involve manipulating the ideal gas law to find unknown quantities about a gas under certain conditions. By rearranging the ideal gas law, \(n = \frac{PV}{RT}\), we can solve for the needed variable. For example, to solve for the number of moles (\(n\)), we need to know the values of pressure (\(P\)), volume (\(V\)), and temperature (\(T\)). We also use the constant \(R\) which is the gas constant.

By inserting the experimental values into the equation, we account for real-life conditions like temperature in Celsius, which we convert to Kelvin, and pressure in units that we adapt to atm. Our calculations directly lead us to the amount of substance we are investigating.

Stoichiometry is the branch of chemistry that deals with the quantitative relationships between the reactants and products in a chemical reaction. It's based on the law of conservation of mass where the total mass of reactants equals the total mass of products.

In electrochemical reactions, stoichiometry tells us how many moles of electrons are required to produce a certain amount of gas. For instance, the stoichiometry of the given half-reaction of water electrolysis indicates that two electrons are needed to produce one molecule of hydrogen gas. Consequently, knowing the amount of hydrogen gas produced enables us to calculate the moles of electrons that have passed through the solution. This understanding is central for balancing chemical equations and for scaling reactions for practical applications such as industrial electrolysis processes.