Calculate the pressure of \(\mathrm{H}_{2}\) (in atm) required to maintain equilibrium with respect to the following reaction at \(25^{\circ} \mathrm{C}\) : $$ \mathrm{Pb}(s)+2 \mathrm{H}^{+}(a q) \rightleftharpoons \mathrm{Pb}^{2+}(a q)+\mathrm{H}_{2}(g) $$ Given that \(\left[\mathrm{Pb}^{2+}\right]=0.035 M\) and the solution is buffered at \(\mathrm{pH} 1.60\).

Understanding the equilibrium constant (K) is key to mastering chemical equilibrium calculations. To imagine what equilibrium means, picture a scale in perfect balance—that is essentially what's happening at the molecular level in a chemical reaction at equilibrium.

The equilibrium constant is a ratio of the concentration of the products to the concentration of the reactants, raised to the power of their stoichiometric coefficients. In the provided exercise, the equilibrium constant expression for the reaction \(\mathrm{Pb}(s) + 2 \mathrm{H}^{+}(aq) \rightleftharpoons \mathrm{Pb}^{2+}(aq) + \mathrm{H}_{2}(g)\) would exclude the solid lead (\(\mathrm{Pb}\) ) because its concentration remains constant. So, the expression simplifies to \(K = \frac{[\mathrm{Pb}^{2+}]}{[\mathrm{H}^{+}]^2}\).

By calculating K, you can predict the direction of the reaction—whether it will go forward or reverse to reach equilibrium. A higher K value indicates that at equilibrium, the reaction favors the formation of products, while a lower K value suggests that reactants are favored.

The pH of a solution is a measure of how acidic or basic that solution is and is defined as the negative logarithm of the hydrogen ion concentration: \(\mathrm{pH} = -\log [\mathrm{H}^{+}]\).

In our exercise, the pH is given as 1.60; a lower pH value indicates a higher concentration of hydrogen ions. This relationship is logarithmic, not linear—meaning each unit change in pH equals a tenfold change in hydrogen ion concentration. For example, a pH of 3 is ten times more acidic than a pH of 4.

Understanding the link between pH and \([\mathrm{H}^{+}]\) is not just important for solving equilibrium problems, but is also essential in fields such as biochemistry, environmental science, and medicine, where precise pH control is vital.

The ideal gas law is a critical equation in chemistry and physics which describes the behavior of an ideal gas. It is often written as \(PV = nRT\), where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin.

From manipulating the ideal gas law, we can calculate the pressure exerted by a gas if the number of moles and the temperature are known. In the given solution, the number of moles of \(\mathrm{H}_{2}\) was derived from its equilibrium concentration, which was found using the equilibrium constant and the concentrations of \(\mathrm{Pb}^{2+}\) and \(\mathrm{H}^{+}\).

It's crucial to always use Kelvin for the temperature in gas law calculations, as it is the standard scientific unit for thermodynamic temperature scale, and remember that the values for R may vary depending on the units of pressure and volume used.