A solution is made up by mixing \(125 \mathrm{~mL}\) of \(0.100 \mathrm{M} \mathrm{AuNO}_{3}\) and \(225 \mathrm{~mL}\) of \(0.049 \mathrm{M} \mathrm{AgNO}_{3}\). Twenty-five mL of a \(0.0100 \mathrm{M}\) solution of \(\mathrm{HCl}\) is then added. \(K_{\text {sp }}\) of \(\mathrm{AuCl}=2.0 \times 10^{-13} .\) When equilibrium is established, will there be- -no precipitate? -a precipitate of \(\mathrm{AuCl}\) only? -a precipitate of \(\mathrm{AgCl}\) only? -a precipitate of both \(\mathrm{AgCl}\) and \(\mathrm{AuCl}\) ?

Understanding the concentration of chemical solutions is crucial when dealing with reactions in a solution. This value tells us how much of a particular substance is present in a certain volume of solvent. It's typically expressed in molarity (M), which is the number of moles of solute per liter of solution.

For example, when different solutions are mixed, the concentration of each solute changes because the total volume of the solution increases. The concentration of a solute is found using the formula:
\[ \text{Concentration (M)} = \frac{\text{Moles of solute}}{\text{Total volume of solution (L)}} \].

This calculation is vital in predicting whether a reaction will proceed to form a precipitate, as seen in the original exercise where the concentrations of AuNO3 and AgNO3 were altered upon mixing.

The reaction quotient, Q, is a measure that compares the concentrations of reactants and products at any point during a reaction to determine which way the reaction will shift to reach equilibrium. It is expressed in the same form as the equilibrium constant but is applicable at any stage of the reaction, not just at equilibrium.

The formula for Q looks like this:
\[ Q = \frac{\text{[products]}}{\text{[reactants]}} \]
where [products] and [reactants] are the molar concentrations of the products and reactants raised to the power of their coefficients from the balanced chemical equation. If Q is less than the equilibrium constant (K), the forward reaction is favored to form more products. Conversely, if Q is greater than K, the reaction will shift to form more reactants.

The solubility product constant, or Ksp, represents the extent to which a compound can dissolve in water. It is specific for a particular ionic compound at a given temperature and is a type of equilibrium constant for the solubility equilibrium.

The formula to determine Ksp is:
\[ K_{sp} = \text{[cation]}^n \times \text{[anion]}^m\]
where [cation] and [anion] are the concentrations of the ions and n and m are their respective stoichiometric coefficients.

Crucially, if the reaction quotient Q is greater than Ksp, the solution is oversaturated, and precipitation will occur to reduce concentrations and reach equilibrium. This is integral to the exercise at hand since Ksp helps us determine whether a precipitate will form when two solutions are mixed.

Precipitation reactions involve the formation of an insoluble solid, known as a precipitate, from the reaction of two soluble salts in solution. When the product of the concentrations of the ions exceeds the solubility product (Ksp) for the resultant compound, a solid will form.

In the context of the exercise, the mixing of AuNO3 and AgNO3 solutions with HCl leads to the possibility of forming AuCl and AgCl precipitates. By calculating Q and comparing it with the known Ksp values, we can predict the formation of a solid. If Q exceeds Ksp, as we saw with AuCl, a precipitate will definitely form.