Question

A concentration of 10-100 parts per billion (by mass) of ${\mathrm{Ag}}^4$ is an effective disinfectant in swimming pools. However, if the concentration exceeds this range, the ${\mathrm{Ag}}^{+}$can cause adverse health effects. One way to maintain an appropriate concentration of ${\mathrm{Ag}}^{+}$is to add a slightly soluble salt to the pool. Using $K_{sp}$ values from Appendix D, calculate the equilibrium concentration of ${\mathrm{Ag}}^{+}$in parts per billion that would exist in equilibrium with (a) AgCl, (b) AgBr, (c) AgI.

Short answer

The equilibrium concentrations of Ag${}^{+}$ ions in parts per billion for AgCl, AgBr, and AgI are 14.25 ppb, 3.21 ppb, and 0.099 ppb, respectively.

Step-by-step solution

1. Write the chemical reaction and the solubility product expression.

The chemical reactions for the dissolution of the three silver halides are: (a) AgCl(s) $\rightleftharpoons$ Ag${}^{+}$(aq) + Cl${}^{-}$ (aq) (b) AgBr(s) $\rightleftharpoons$ Ag${}^{+}$(aq) + Br${}^{-}$ (aq) (c) AgI(s) $\rightleftharpoons$ Ag${}^{+}$(aq) + I${}^{-}$ (aq) The solubility product expression for each reaction would be: (a) $K_{sp}=[A{g}^{+}][C{l}^{-}]$ (b) $K_{sp}=[A{g}^{+}][B{r}^{-}]$ (c) $K_{sp}=[A{g}^{+}][I^{-}]$ Since the salts are only slightly soluble, we can assume that the concentration of Cl${}^{-}$, Br${}^{-}$, I${}^{-}$ is equal to the concentration of silver ions [Ag${}^{+}$] to maintain electroneutrality.

2. Calculate the equilibrium concentration of Ag${}^{+}$.

The solubility product expression for each reaction becomes: (a) $K_{sp}=[A{g}^{+}{]}^2$ for AgCl (b) $K_{sp}=[A{g}^{+}{]}^2$ for AgBr (c) $K_{sp}=[A{g}^{+}{]}^2$ for AgI Solving for the concentration of Ag${}^{+}$, we get: (a) ${\text{[Ag}}^{+}]=\sqrt{K_{sp}}$ for AgCl (b) ${\text{[Ag}}^{+}]=\sqrt{K_{sp}}$ for AgBr (c) ${\text{[Ag}}^{+}]=\sqrt{K_{sp}}$ for AgI Using the known $K_{sp}$ values (from Appendix D) for each salt: (a) ${\text{[Ag}}^{+}]=\sqrt{1.8\times {10}^{-10}}$ for AgCl (b) ${\text{[Ag}}^{+}]=\sqrt{5.0\times {10}^{-13}}$ for AgBr (c) ${\text{[Ag}}^{+}]=\sqrt{8.5\times {10}^{-17}}$ for AgI

3. Convert the concentrations to ppb.

To convert the equilibrium concentrations of silver ions (Ag${}^{+}$) from molar units to parts per billion (ppb), multiply the molar concentration by the molar mass of silver (107.87 g/mol) and divide by the sample mass (10${}^9$ g of water for 1 ppb): (a) $pp{b}_{AgCl}=\frac{(\sqrt{1.8\times {10}^{-10}})(107.87)}{{10}^{-9}}$ (b) $pp{b}_{AgBr}=\frac{(\sqrt{5.0\times {10}^{-13}})(107.87)}{{10}^{-9}}$ (c) $pp{b}_{AgI}=\frac{(\sqrt{8.5\times {10}^{-17}})(107.87)}{{10}^{-9}}$ Calculate the ppb values: (a) $pp{b}_{AgCl}\approx 14.25$ ppb (b) $pp{b}_{AgBr}\approx 3.21$ ppb (c) $pp{b}_{AgI}\approx 0.099$ ppb Thus, the equilibrium concentrations of Ag${}^{+}$ ions in parts per billion for AgCl, AgBr, and AgI are 14.25 ppb, 3.21 ppb, and 0.099 ppb, respectively.

Key concepts

Equilibrium Concentration

When working with slightly soluble salts in a solution, we often encounter the concept of equilibrium concentration. This refers to the steady-state concentration of ions in a solution when a solute is in dynamic equilibrium with its dissolved ions. In this exercise, we are interested in the equilibrium concentration of silver ions, [Ag${}^{+}$], in the presence of different silver halides like AgCl, AgBr, and AgI.

To find this equilibrium concentration, we make use of the solubility product constant ($K_{sp}$). $K_{sp}$ essentially tells us how much of the salt can dissolve in water at a given temperature. For instance, the equation for AgCl(s) dissolving in water is: AgCl(s) $\rightleftharpoons$ Ag${}^{+}$(aq) + Cl${}^{-}$ (aq).
The solubility product expression for this reaction is $K_{sp}=[Ag{\text{(}}^{+}$][Cl${}^{-}$]\). When the solution reaches equilibrium, the concentrations of [Ag${}^{+}$] and [Cl${}^{-}$] are equal (let's call it $[Ag{\text{(}}^{+}$] = x\)). From there, we can simplify the expression to $K_{sp}=x^2$. By taking the square root of $K_{sp}$, we can solve for $x$, which represents the equilibrium concentration of [Ag${}^{+}$].

Understanding this concept helps us predict how much of a salt will dissolve in a solution, maintaining the desired concentration for specific applications, such as keeping silver ion levels within safe ranges in swimming pools.

Chemical Reactions

The chemical reactions involved in dissolving slightly soluble salts like silver halides are fundamental to understanding the broader context of solubility equilibria. Each silver halide salt, such as AgCl, AgBr, and AgI, behaves slightly differently due to the varying strength of their ionic bonds.

The dissolution process can be represented by simple chemical equations:

• For AgCl: AgCl(s) $\rightleftharpoons$ Ag${}^{+}$(aq) + Cl${}^{-}$ (aq)
• For AgBr: AgBr(s) $\rightleftharpoons$ Ag${}^{+}$(aq) + Br${}^{-}$ (aq)
• For AgI: AgI(s) $\rightleftharpoons$ Ag${}^{+}$(aq) + I${}^{-}$ (aq)

In each of these reactions, the solid salt shifts to form ions in a solution. This shift represents a dynamic equilibrium where the rate of dissolution equals the rate of precipitation of ions back to form the solid salt. Understanding these reactions is crucial because they allow us to predict the solubility of the salt in water, which is inherently tied to the salt's $K_{sp}$ value. These underlying chemical reactions provide the foundation for calculating solubility and concentrations, helping to control chemical processes effectively.

Parts Per Billion

When dealing with the concentration of substances like metals in large volumes of water, we often express these concentrations in parts per billion (ppb). Understanding and using parts per billion is crucial in fields like chemistry and environmental science because it allows us to measure extremely dilute concentrations.

The term "ppb" essentially means "out of a billion." It represents the ratio of the mass of the solute to the mass of the solution, multiplied by 10${}^9$. This unit is especially useful when measuring trace amounts of pollutants or additives in water solutions, such as in swimming pools.

To convert a concentration from molarity to ppb, we use the formula:$$ppb=\left(\text{molar concentration}\right)\times \left(\text{molar mass of the substance}\right)÷10{\text{(}}^9\text{)}$$This calculation helps visualize how much of a substance, in this case, silver ions, is present in water relative to a billion parts of water.

Recognizing how to measure in parts per billion can help ensure compliance with safety standards and health guidelines, preventing situations where toxic concentrations inadvertently occur.