Finding solubility by iteration. Use the systematic treatment of equilibrium
to find the concentrations of the major species in a saturated aqueous
solution of LiF. Consider these reactions:
$$
\begin{array}{ll}
\mathrm{LiF}(s) \rightleftharpoons \mathrm{Li}^{+}+\mathrm{F}^{-} & K_{\text
{sp }}=\left[\mathrm{Li}^{+}\right]
\gamma_{\mathrm{Li}^{-}}\left[\mathrm{F}^{-}\right]
\gamma_{\mathrm{F}^{-}}=0.0017 \\
\mathrm{LiF}(s) \rightleftharpoons \mathrm{LiF}(a q) & K_{\text {ion pair
}}=\mathrm{LiF}(a q) \gamma_{\mathrm{LiF}(a q)}=0.0029
\end{array}
$$
(a) Initially, set the ionic strength to 0 and solve for all concentrations.
Then compute the ionic strength and activity coefficients and find new
concentrations. Use several iterations to home in on the correct solution.
(b) In the systematic treatment, you will find that the calculation simplifies
to
\(\left[\mathrm{Li}^{+}\right]=\left[\mathrm{F}^{-}\right]=\sqrt{K_{\mathrm{sp}}
/\left(\gamma_{\mathrm{Li}^{+}} \gamma_{\mathrm{F}^{-}}\right)}\). Set up the
following spreadsheet, in which ionic strength in cell B4 is initially given
the value 0 . Activity coefficients in cells B6 and B8 are from the extended
Debye-H眉ckel equation. Cell B10 computes
\(\left[\mathrm{Li}^{+}\right]=\left[\mathrm{F}^{-}\right]=\sqrt{K_{\mathrm{sp}}
/\left(\gamma_{\mathrm{Li}^{+}} \gamma_{\mathrm{F}}\right) . \text { With } 0
\text { in cell B4, your spread- }}\) sheet should compute 1 in cells
\(\mathrm{B} 6\) and \(\mathrm{B} 8\) and
\(\left[\mathrm{Li}^{+}\right]=\left[\mathrm{F}^{-}\right]=\) \(0.04123
\mathrm{M}\) in cell \(\mathrm{B} 10\).
Because the ionic strength of a \(1: 1\) electrolyte is equal to the
concentration, copy the value \(0.04123\) from cell B10 into cell B4. (To
transfer a numerical value, rather than a formula, copy cell B 10 and then
highlight cell B4. In Excel 2007, go to the Home ribbon, select Paste, and
then click on Paste Values. The numerical value from \(\mathrm{B} 10\) will be
pasted into \(\mathrm{B} 4\). In earlier versions of Excel, go to the Edit menu,
select Paste Special, and then choose Value.) This procedure gives new
activity coefficients in cells B6 and B8 and a new concentration in cell B10.
Copy the new concentration from cell B 10 into cell B4 and repeat this
procedure several times until you have a constant answer.
(c) Using circular references in Excel. Take your spreadsheet and enter 0 in
cell B4. The value \(0.04123\) is computed in cell B10. Ideally, you would like
to write " \(=\mathrm{B} 10 "\) in cell \(\mathrm{B} 4\) so that the value from
B10 would be copied to B4. Excel gives a "circular reference" error message
because cell \(\mathrm{B} 10\) depends on \(\mathrm{B} 4\) and cell \(\mathrm{B} 4\)
depends on B10. To get around this problem, in Excel 2007, click the Microsoft
Office button at the upper left of the spreadsheet. Click on Excel Options at
the bottom of the window. Select Formulas. In Calculation Options, check
Enable iterative calculation and set Maximum Change to \(0.00001\). Click
\(\mathrm{OK}\) and Excel merrily iterates between cells \(\mathrm{B} 10\) and
\(\mathrm{B} 4\) until the two values agree within \(0.00001\). In earlier
versions of Excel, go to the Tools menu and choose Options. Select Calculation
and choose Iteration. Set Maximum change to \(0.00001\).
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