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Chapter 39: Problem 10

A solution of lithium carbonate, \(\mathrm{Li}_{2} \mathrm{CO}_{3}\), reaches equilibrium with its solid after \(0.0742\) moles of the solid have dissolved in \(1.00\) liter. A student suggests that the value of \(K_{\mathrm{sp}}\) for lithium carbonate is \((0.0742)^{2}(0.0742) .\) Explain why the student is not correct and then calculate the correct value of \(K_{\mathrm{sp}}\) for lithium carbonate.

Short Answer

Expert verified
The student's calculation is incorrect because they failed to account for the correct stoichiometry, where for each mole of \(\mathrm{Li}_{2} \mathrm{CO}_{3}\) that dissolves, two moles of \(\mathrm{Li}^{+}\) are produced. Thus, the correct calculation for \(K_{sp}\) should be \((0.1484)^{2} \times 0.0742\).

Step by step solution

01

Writing the Equilibrium

Firstly, we must write down the equilibrium for the dissolution of lithium carbonate in water. The balanced chemical equation for this process is: \(\mathrm{Li}_{2} \mathrm{CO}_{3(s)} \rightleftharpoons 2 \mathrm{Li}^{+}_{(aq)} + \mathrm{CO}_{3}^{2-}_{(aq)}\). This equation shows that for every one mole of \(\mathrm{Li}_{2} \mathrm{CO}_{3}\) that dissolves, two moles of \(\mathrm{Li}^{+}\) ions and one mole of \(\mathrm{CO}_{3}^{2-}\) ion are produced in the solution.
02

Identifying the Error

The student incorrectly assumed the concentrations of \(\mathrm{Li}^{+}\) and \(\mathrm{CO}_{3}^{2-}\) to be equal (hence cubing 0.0742) because one mole of \(\mathrm{Li}_{2} \mathrm{CO}_{3}\) was dissolved. However, referring back to the equilibrium equation, for every mole of \(\mathrm{Li}_{2} \mathrm{CO}_{3}\) that dissolves, TWO moles of \(\mathrm{Li}^{+}\) are produced. Therefore, the concentration of \(\mathrm{Li}^{+}\) is twice that of \(\mathrm{CO}_{3}^{2-}\) upon dissolution.
03

Calculating Correct \(K_{sp}\) Value

The \(K_{sp}\) is calculated from the equilibrium concentrations of the ions. It's formula from the balanced chemical equation is: \(K_{sp} = [\mathrm{Li}^{+}]^{2}[\mathrm{CO}_{3}^{2-}]\). Given that 0.0742 moles of \(\mathrm{Li}_{2}CO_{3}\) are dissolved in 1.00 litre, the concentration of \(\mathrm{CO}_{3}^{2-}\) is 0.0742 M and the concentration of \(\mathrm{Li}^{+}\) is 2*0.0742 M = 0.1484 M. Hence, the correct \(K_{sp}\) value is: \(K_{sp} = (0.1484)^{2}(0.0742)\).

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Most popular questions from this chapter

Is \(\mathrm{Mg}(\mathrm{OH})_{2}\) considered to be a soluble or insoluble compound? Explain how you reached this conclusion.

A student adds \(0.07\) moles of an ionic compound to \(500 \mathrm{~mL}\) of water and all of the material dissolves. Is the ionic compound soluble, insoluble, moderately soluble, or is it impossible to tell? Explain your reasoning and include at least one calculation in your analysis.

Write the solubility constant expression, \(K_{\mathrm{sp}}\), for each reaction below. a) \(\mathrm{AgCl}(\mathrm{s}) \rightleftarrows \mathrm{Ag}^{+}(\mathrm{aq})+\mathrm{Cl}^{-}(\mathrm{aq})\) b) \(\mathrm{Cu}_{3}\left(\mathrm{PO}_{4}\right)_{2}(\mathrm{~s}) \rightleftarrows 3 \mathrm{Cu}^{2+}(\mathrm{aq})+2 \mathrm{PO}_{4}^{3-}(\mathrm{aq})\) c) \(\mathrm{Li}_{2} \mathrm{CO}_{3}(\mathrm{~s}) \rightleftarrows 2 \mathrm{Li}^{+}(\mathrm{aq})+\mathrm{CO}_{3}^{2-(\mathrm{aq})}\)
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