Chapter 17: Problem 83
At \(298 \mathrm{K}, K_{\mathrm{sp}}\) for cadmium iodate \(\left(\mathrm{Cd}\left(\mathrm{IO}_{3}\right)_{2}\right)\) equals \(2.3 \times 10^{-8} .\) What are the molar concentrations of cadmium ions and iodate ions in a saturated solution at 298 \(\mathrm{K?}\)
Short Answer
Expert verified
The concentration of \(\mathrm{Cd}^{2+}\) is \(1.77 \times 10^{-3}\, \mathrm{mol/L}\) and \(\mathrm{IO}_3^-\) is \(3.54 \times 10^{-3}\, \mathrm{mol/L}\).
Step by step solution
01
Understanding the Dissolution Process
Cadmium iodate, \( \mathrm{Cd}\left(\mathrm{IO}_{3}\right)_{2} \), dissociates in water to form one cadmium ion (\( \mathrm{Cd}^{2+} \)) and two iodate ions (\( \mathrm{IO}_3^- \)). This can be expressed as the equation: \( \mathrm{Cd}\left(\mathrm{IO}_{3}\right)_{2} (s) \rightleftharpoons \mathrm{Cd}^{2+} (aq) + 2 \mathrm{IO}_3^- (aq) \).
02
Writing the Expression for Ksp
The solubility product constant \( K_{\mathrm{sp}} \) is given by the product of the concentrations of the ions raised to the power of their stoichiometric coefficients: \( K_{\mathrm{sp}} = [\mathrm{Cd}^{2+}][\mathrm{IO}_3^-]^2 \). Here, \( [\mathrm{Cd}^{2+}] = x \) and \( [\mathrm{IO}_3^-] = 2x \).
03
Substituting into the Ksp Expression
Substitute the expressions for the ion concentrations into the \( K_{\mathrm{sp}} \) equation: \( K_{\mathrm{sp}} = x(2x)^2 = 4x^3 \).
04
Solving for x
With \( K_{\mathrm{sp}} = 2.3 \times 10^{-8} \), set \( 4x^3 = 2.3 \times 10^{-8} \). Solve for \( x \) by dividing both sides by 4: \( x^3 = \frac{2.3 \times 10^{-8}}{4} = 5.75 \times 10^{-9} \). Then take the cube root of both sides to find \( x \): \( x = \sqrt[3]{5.75 \times 10^{-9}} \).
05
Calculating x and Ion Concentrations
Calculate \( x = \sqrt[3]{5.75 \times 10^{-9}} \approx 1.77 \times 10^{-3} \). This is the concentration of \( \mathrm{Cd}^{2+} \). The concentration of \( \mathrm{IO}_3^- \) is \( 2x \) or \( 2 \times 1.77 \times 10^{-3} = 3.54 \times 10^{-3} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Dissolution Process
In the dissolution process, cadmium iodate (\(\mathrm{Cd}\left(\mathrm{IO}_{3}\right)_{2}\))dissolves in water to separate into ions. This process helps us understand how compounds break down in aqueous solutions. When cadmium iodate dissolves, it splits into cadmium ions (\(\mathrm{Cd}^{2+}\))and iodate ions (\(\mathrm{IO}_3^-\)).The reaction can be shown as:- \[\mathrm{Cd}\left(\mathrm{IO}_{3}\right)_{2} (s) \rightleftharpoons \mathrm{Cd}^{2+} (aq) + 2 \mathrm{IO}_3^- (aq)\]This means every formula unit of cadmium iodate produces one cadmium ion and two iodate ions when dissolved in water. The process is reversible, as indicated by the double-headed arrow, which means at equilibrium, cadmium iodate is consistently dissolving and precipitating back into a solid at equal rates.
Molar Concentration
Molar concentration measures how many moles of a substance are in one liter of solution. It is crucial for determining the concentration of ions produced in a dissolution process. For instance, in a saturated solution of cadmium iodate, knowing the molar concentration tells us the amount of cadmium and iodate ions present.In our case, when cadmium iodate dissolves:- Cadmium ion concentration = \([\mathrm{Cd}^{2+}] = x\)- Iodate ion concentration = \([\mathrm{IO}_3^-] = 2x\)Here, \(x\) represents the molar concentration of cadmium ions, and \(2x\) is for iodate ions, as each unit of \(\mathrm{Cd}\left(\mathrm{IO}_{3}\right)_{2}\) produces two iodate ions. Calculating these concentrations helps us understand the solubility effectiveness of the compound.
Cadmium Iodate
Cadmium iodate, (\(\mathrm{Cd}\left(\mathrm{IO}_{3}\right)_{2}\)), is a compound that is sparingly soluble in water. This means it dissolves to a small extent and creates an equilibrium between its dissolved ions and the remaining solid. The Ksp (solubility product constant) provides a way to quantify this solubility.Cadmium iodate is noteworthy for reactions involving precipitation and demonstrating ionic equilibrium in solutions. At 298 K, its Ksp value of \(2.3 \times 10^{-8}\) suggests limited solubility. This is crucial for experiments and calculations that involve determining the concentration of resulting ions once the compound dissolves.
Equilibrium Expression
The equilibrium expression is a mathematical representation of the ratio of the concentrations of products to reactants at equilibrium. It depends on the principle of the solubility product constant (\(K_{sp}\)), which simplifies the calculation of ion concentrations in a saturated solution.For cadmium iodate:- The equation is (\(\mathrm{Cd}\left(\mathrm{IO}_{3}\right)_{2} (s) \rightleftharpoons \mathrm{Cd}^{2+} (aq) + 2 \mathrm{IO}_3^- (aq)\)).- The Ksp is given by: \[K_{sp} = [\mathrm{Cd}^{2+}][\mathrm{IO}_3^-]^2\]If we replace:[\mathrm{Cd}^{2+}] with (\(x\)) and [\mathrm{IO}_3^-] with (\(2x\)),We derive:- \[K_{sp} = x(2x)^2 = 4x^3\]This expression is key to solving ion concentration problems, as it helps in calculating x, which represents the molar concentration of dissolved ions at equilibrium.