Solubility of Ag3AsO4 in 0.02 M K3AsO4

Question

Calculate the solubility of $\ce{Ag3AsO4}$ in $\pu{0.02M}~\ce{K3AsO4}$ neglecting the activity coefficients. Find the relative error. $K_\mathrm{sp}(\ce{Ag3AsO4}) = \pu{6.0e-23}$

I know how to calculate the relative error but I get a very complicated equation finding the concentration solubility product constant ($K'_\mathrm{sp}$). There should be a quicker way to solve this since it is a midterm question.

I tried this:

$$\ce{Ag3AsO4 -> 3Ag+ + AsO4^3-}$$

$$K_\mathrm{sp} = 27x^4$$

$$x = \pu{2.78e-6}$$

$$[\ce{Ag+}] = 3x = \pu{8.34e-6M}$$

$$[\ce{AsO4^3-}] = x = \pu{2.78e-6M}$$

$0.02~\mathrm{M}\ \ce{AsO4^3-}$ comes from $\ce{K2AsO4}$. So there should be an equation like:

$$K_\mathrm{sp} = (\pu{8.34e-6} - 3x)^3 \times (\pu{2.78e-6} + 0.02 - x) = \pu{6.0e-6}$$

And things get complicated. After finding $x$, I will also have found the final concentrations of silver and $\ce{AsO4^3-}$ ions. Then I will read the activity coefficients of them from the appendix table.

I have two questions:

1. Is my method true or false?

2. What is an easier way of solving this problem?

Answer 1

Your calculations are mathematically correct up to the second step and then I have no clue how you arrived at your value of $x$. Starting from there, $x$ would be:

$$K_\mathrm{sp} = 27 x^4 \\ \frac{K_\mathrm{sp}}{27} = x^4 \\ \sqrt[4]{\frac{K_\mathrm{sp}}{27}} = x \\ x \approx 0.021711\dots$$

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However, to calculate the solubility you should be doing this:

$$K_\mathrm{sp} = \left[\ce{Ag+}\right]^3 \left[\ce{AsO4^3-}\right]$$

$$c\left (\ce{Ag3AsO4} \right) = x$$

$$K_\mathrm{sp} = \left(3x \right )^3 \left(x + 0.02~\mathrm{M}\right)\\ K_\mathrm{sp} = 27x^4 + 0.54~\mathrm{M} \cdot x^3$$

Note that this is the same equation as Uros proposed. I arrived there by saying:

1. $\left [\ce{Ag+}\right] = 3\ c\left(\ce{Ag3AsO4}\right)$ — silver ions all come from dissolved $\ce{Ag3AsO4}$

2. $\left [\ce{AsO4^3-}\right] = c\left(\ce{Ag3AsO4}\right) + 0.02~\mathrm{M}$ — $0.02~\mathrm{M}$ of $\ce{AsO4^3-}$ stem from $\ce{K3AsO4}$, the remaining from dissolved $\ce{Ag3AsO4}$.

Unfortunately, our equation is not (easily) solveable analytically to the best of my knowledge. You need some kind of estimation method. Uros’ estimate of $c\left(\ce{Ag3AsO4}\right) \approx 0.018~\mathrm{M}$ seems pretty accurate.

Answer 2

You start of by calculating solubility with one approximation: that all $\ce{AsO4^3-}$ ions are obtained by dissolving the potassium salt:

Let $S$ be the molar solubility:

$$K_\mathrm{sp} = 0.02 \times 27 S^3$$

$$S = \pu{0.0223 M}$$

Then, you calculate the solubility by taking in account the concentration of arsenate ion obtained by dissolution of silver arsenate:

$$K_\mathrm{sp} = 27 S^3 \times (S+0.02) = 27 S^4 + 0.54S^3$$

Solving this equation by applying the iteration method gives the value of $S = \pu{0.018M}$.

So the relative error is $23.9~\%$.

Answer 3

The correct solution:

$$K_\mathrm{sp} = a_\mathrm{Ag^+}^3 \cdot a_\mathrm{AsO_{4}^{3-}} = 6.0 \cdot 10^{-23}$$

$$K_\mathrm{sp} = K_\mathrm{sp}^{'} \cdot γ_\mathrm{Ag^+}^3 \cdot γ_\mathrm{AsO_{4}^{3-}}$$

$$K_\mathrm{sp}^{'} = \frac{K_\mathrm{sp}}{γ_\mathrm{Ag^+}^3 \cdot γ_\mathrm{AsO_{4}^{3-}}}$$

Ionic strength:

$$μ = \frac{1}{2} \cdot (0,06\cdot1^2 + 0,02\cdot3^2) = 0,12~\mathrm{M}$$

$$0,12~\mathrm{M} ≈ 0,10~\mathrm{M} $$

Activity coefficients at ionic strength $0,12~\mathrm{M}$ are:

$0.75$ for $Ag^+$

$x$ for $AsO_{4}^{3-}$

Then:

$$K_\mathrm{sp}^{'} = \frac{K_\mathrm{sp}}{0.75^3 \cdot x}$$

And the relative error is: $$\%~error = \frac{|K_\mathrm{sp}-K_\mathrm{sp}^{'}|}{K_\mathrm{sp}} \cdot 100~\% = \frac{|6.0 \cdot 10^{-23}-K_\mathrm{sp}^{'}|}{6.0 \cdot 10^{-23}} \cdot 100~\%$$

I can't find the activity coefficient for arsenate. If someone can find it, please share it with us.