A mixture of \(\mathrm{H}_{2}, \mathrm{~S},\) and \(\mathrm{H}_{2} \mathrm{~S}\) is held in a \(1.0-\mathrm{L}\) vessel at \(90{ }^{\circ} \mathrm{C}\) and reacts according to the equation: $$\mathrm{H}_{2}(g)+\mathrm{S}(s) \rightleftharpoons \mathrm{H}_{2} \mathrm{~S}(g)$$ At equilibrium the mixture contains \(0.46 \mathrm{~g}\) of \(\mathrm{H}_{2} \mathrm{~S}\) and \(0.40 \mathrm{~g}\) \(\mathrm{H}_{2}\) (a) Write the equilibrium-constant expression for this reaction. (b) What is the value of \(K_{c}\) for the reaction at this temperature? (c) Why can we ignore the amount of \(\mathrm{S}\) when doing the calculation in part (b)?

Chemical equilibrium is a fundamental concept in chemistry that occurs when the rate of the forward reaction equals the rate of the reverse reaction, and the concentrations of the reactants and products remain constant over time. It's important to realize that this doesn't mean the reactions stop; both reactions still occur, but at the same rate, leading to a stable ratio of products and reactants.

In the context of our exercise, the reaction between hydrogen gas (\textbf{H}\(_2\)) and sulfur (\textbf{S}) to form hydrogen sulfide (\textbf{H}\(_2\)\textbf{S}) reaches chemical equilibrium in a closed vessel. At equilibrium, while the amounts of the reactants and products stay constant, it doesn't indicate that they are in equal concentrations. The equilibrium constant (\textbf{K}\(_c\)) expresses the ratio of these concentrations at equilibrium, providing a quantitative measure of the system's balance.

To accurately determine the equilibrium constant, a key step is to convert the mass of substances involved in the reaction to moles, since \textbf{K}\(_c\) is based on molar concentrations.To calculate molar mass, you sum the atomic masses of all atoms in the molecule. For instance, the molar mass of \textbf{H}\(_2\) is 2.016 g/mol (based on the atomic mass of Hydrogen, 1.008 g/mol, multiplied by 2), and for \textbf{H}\(_2\)\textbf{S}, it is 34.08 g/mol. By dividing the given masses by these molar masses, we obtain the number of moles, which are then used to find the concentration by dividing by the volume of the reaction vessel. As seen in our exercise, these calculations are crucial to determine the equilibrium concentrations required for computing the equilibrium constant.

After finding the number of moles of each species, the next step is to calculate their concentrations. This is done by dividing the number of moles by the volume of the vessel, assuming ideal behavior and that the gaseous volume occupies the entire container. In our exercise, the reaction reaches equilibrium within a 1.0 L vessel, allowing for direct translation from moles to molar concentration (\textbf{M}), since concentration is defined as moles per liter (\textbf{mol/L}).

For a balanced chemical system like the one described in the exercise, these equilibrium concentrations enable us to compute the equilibrium constant (\textbf{K}\(_c\)) that is specific to the temperature of the system. It's worth noting that \textbf{K}\(_c\) is dimensionless and can vary widely in magnitude, reflecting the extent to which a reaction goes to completion under a given set of conditions.