Determining solubility of silver sulfate in its saturated solution

Question

Question:

Calculate the molar sulfate concentration in a solution that is $0.200$ M in $\ce{Ag+}$ and saturated with $\ce{Ag2SO4}$.

My attempt:

Looking through old notes and googling lead me to this webpage. Example #2 looked similar to my problem, and my notes from general chemistry involved something called an ICE table to deal with stoichiometry and concentrations of ions. This is what I did:

1. Write the dissociation equation $\ce{Ag2SO4 + <=> 2Ag+ + SO4^2-}$
2. Write the solubility constant formula $$K_\mathrm{sp} = \frac{\text{products}}{\text{reactants}}$$

I was given the value of $K_\mathrm{sp}$ as $1.6 \times 10^{-5}$ and I assume these values are listed in an appendix in my text or online.

Also $\ce{Ag2SO4}$ is supposed to be a solid, I think? Because the question states the solution is "saturated" with $\ce{Ag2SO4}$ and does "saturation" not imply that the $\ce{Ag2SO4}$ is going in and out of solution as a solid and as an ion? And therefore since it is a solid we do not consider it for the equation (not sure why, but this is similar to what I have in my notes)?

$$1.6 \times 10^{-5} = \text{products}$$

$$1.6 \times 10^{-5} = \ce{[Ag+]^2 * [SO4^2- ]}$$

Here is an image if that is acceptable of the ICE table I used (note: $\text{I+C=E}$).

$$\begin{array}{|c|c|c|}\hline &\ce{Ag+}&\ce{SO4^2-}\\\hline \text{I}&\pu{0.200M}&x\\\hline \text{C}&\frac x2&0\\\hline \text{E}&\pu{0.200M}+\frac x2&x\\\hline \end{array}$$

Now I substitute the values I obtained for concentrations from the ICE table into my formula:

$$1.6 \times 10^{-5} = [0.200~\text{M} + \frac{1}{2}x]^2 \times [x]$$

Now since $K_\mathrm{sp}$ is less than $1 \times 10^{-5}$ we can ignore the $\frac{1}{2}x$? I guess because it is very small and therefore negligible?

$$1.6 \times 10^{-5} = [0.200~\text{M}]^2 \times [x]$$

$$ \frac{1.6 \times 10^{-5}}{[0.200~\text{M}]^2} = x = 0.0004$$

$$ x = 4.0 \times 10^{-4}~\text{M}$$

Therefore the concentration of $\ce{SO4^2-}$ is $ 4.0 \times 10^{-4}~\text{M}$?

Answer 1

Solubility constants

You correctly state that they are defined as:

$$K_\mathrm{sp} = \frac{[\text{products}]}{[\text{rectants}]}$$

And in your case ($\ce{Ag2SO4}$) this would be:

$$K_\mathrm{sp} = \frac{[\ce{Ag+}]^2 \cdot [\ce{SO4^2-}]}{[\ce{Ag2SO4}]}$$

And concerning the reactand, the ‘concentration’ of a solid is always defined as $1$, so it is removed from the equation (technically this is an activity not a concentration but that doesn’t matter in this case).

The problem itself

What do you have? Let’s take another look at it:

Calculate the molar sulfate concentration in a solution that is $0.200\,\mathrm{M}$ in:

a) $\ce{Ag+}$ and saturated with $\ce{Ag2SO4}$

So we have dissolved $\ce{Ag+}$ with a concentration of $[\ce{Ag+}] = 0.200\,\mathrm{M}$ in aquaeous solution and also a solid $\ce{Ag2SO4}$ precipitate. You also state you are given the value of $K_\mathrm{sp} = 1.6 \times 10^{-5}$. And you are supposed to calculate $[\ce{SO4^2-}]$.

You have all the information you need and all you need to do is solve a single equation.