Question

A voltaic cell, with \(E_{\text {cell }}=0.180 \mathrm{V},\) is constructed as follows: $$\mathrm{Ag}(\mathrm{s})\left|\mathrm{Ag}^{+}\left(\operatorname{satd} \mathrm{Ag}_{3} \mathrm{PO}_{4}\right) \| \mathrm{Ag}^{+}(0.140 \mathrm{M})\right| \mathrm{Ag}(\mathrm{s})$$ What is the \(K_{\mathrm{sp}}\) of \(\mathrm{Ag}_{3} \mathrm{PO}_{4} ?\)

Short answer

To solve this problem, first use the Nernst equation to determine the concentration of Ag+ in the solution. Then use this concentration to calculate the solubility product constant (\(K_{\mathrm{sp}}\)) of \(\mathrm{Ag}_3 \mathrm{PO}_4\).

Step-by-step solution

Identify the relevant equations

The Nernst equation is given by: \[E_{\text {cell }}=E^{\circ}-\left(\frac{0.0592}{n}\right) \log Q\]The value of \(Q\) for the reaction is \([Ag^{+}]\). Since there is no information about the standard electrode potential \(E^{\circ}\) the equation simplifies to:\[E_{\text {cell }}=-\left(\frac{0.0592}{n}\right) \log Q\]or \[E_{\text {cell }}=\left(\frac{-0.0592}{n}\right) \log \left(\frac{1}{[Ag^{+}]}\right)\]

Calculate the concentration of Ag+

In the voltaic cell, the reduction potential is given as 0.180 V. Substituting this into the Nernst equation, we get:\[0.180 = \left(\frac{-0.0592}{n}\right) \log \left(\frac{1}{[Ag^{+}]}\right)\]Assuming a single-electron process (n=1), we can solve for [Ag+]. This leads to:\[\log \left(\frac{1}{[Ag^{+}]}\right) = \frac{0.180}{-0.0592}\]Calculate [Ag+] to get:\[[Ag^+] = 0.140 \, M\]

Calculate the Ksp

The equilibrium constant for the solubility of \(\mathrm{Ag}_3 \mathrm{PO}_4\) is:\[K_{\mathrm{sp}} = [Ag^+]^3[PO_4^{3-}]\]For a saturated solution of \(\mathrm{Ag}_3 \mathrm{PO}_4\), [Ag+] is three times the [PO4^3-]. This leads to \[K_{\mathrm{sp}} = [Ag^+]^3 \times \left(\frac{[Ag^+]}{3}\right)\]Substitute [Ag+] with the concentration calculated from Step 2 and solve for \(K_{\mathrm{sp}}\).

Key concepts

Nernst Equation

The Nernst equation is an essential element in understanding electrochemical cells and their potentials. This equation allows us to calculate the cell potential based on the concentrations of the ions involved. It is expressed as follows: \[E_{\text{cell}} = E^{\circ} - \left(\frac{0.0592}{n}\right) \log Q\]Here, \(E^{\circ}\) is the standard cell potential, \(n\) is the number of moles of electrons exchanged in the electrochemical reaction, and \(Q\) is the reaction quotient. In simpler terms, the Nernst Equation helps us understand how changes in concentration can affect the voltage generated by an electrochemical cell.
When we do not have the standard electrode potential \(E^{\circ}\) for a system, or when dealing with a system under non-standard conditions (like different concentrations), the Nernst equation becomes crucial for calculating the cell potential. It adjusts the potential for changes in ion concentrations.
The equation can also be rearranged, like in the problem, to solve for ion concentrations when potential data is available. This is integral in determining the extent of solubility of a species or the strength of an electrochemical cell.
Understanding the Nernst equation helps in configuring and predicting the behavior of real and practical electrochemical systems, such as batteries and sensors.

Solubility Product Constant

Solubility Product Constant, often abbreviated as \(K_{\text{sp}}\), is a vital concept when dealing with the solubility of sparingly soluble salts in solution. It is the product of the concentrations of the ions of a salt, each raised to the power of its stoichiometric coefficient in the balanced equation.
For example, for silver phosphate \(\text{Ag}_3\text{PO}_4\), which dissolves to form silver ions \(\text{Ag}^+\) and phosphate ions \(\text{PO}_4^{3-}\) in water, the \(K_{\text{sp}}\) expression is given by:\[K_{\text{sp}} = [Ag^+]^3[PO_4^{3-}]\]The value of \(K_{\text{sp}}\) provides a measure of the solubility. It signifies how much of the salt can dissolve in water to form a saturated solution. A high \(K_{\text{sp}}\) indicates greater solubility.
In the provided exercise, by linking the concentration of silver ions from the silver phosphate dissolution to the cell potential via the Nernst equation, we can determine \(K_{\text{sp}}\).
This calculation is crucial in analytical chemistry, environmental science, and various fields where solubility products govern the movement and reactivity of substances in liquid environments.

Electrochemistry

Electrochemistry is the branch of chemistry that studies the relationship between electricity and chemical reactions. It encompasses a range of phenomena from batteries to corrosion, to the electrolysis processes used in industry.
In the context of the voltaic cell mentioned, electrochemistry is at play in generating electric current from chemical reactions. Voltaic (or galvanic) cells use spontaneous redox reactions to generate an electric current.
These cells consist of two different electrodes immersed in electrolyte solutions, where oxidation and reduction occur. The two half-reactions are connected by a wire through which electrons can flow, generating a current.

• Anode: The electrode where oxidation occurs. It gives off electrons to the external circuit.
• Cathode: The electrode where reduction takes place. It receives electrons from the circuit.

Understanding electrochemistry is fundamental in designing batteries, understanding corrosion, and even biochemistry processes, where electron transfers are crucial.

Silver Phosphate (Ag3PO4)

Silver Phosphate \(\text{Ag}_3\text{PO}_4\) is a sparingly soluble ionic compound, often notable for its bright yellow color and use in photography and as an antimicrobial agent.
Silver phosphate dissolves in water forming silver ions \(\text{Ag}^+\) and phosphate ions \(\text{PO}_4^{3-}\). This dissolution is characterized by its solubility product \(K_{\text{sp}}\), which helps quantify its solubility in water:\[\text{Ag}_3\text{PO}_4 \rightleftharpoons 3\text{Ag}^+ + \text{PO}_4^{3-}\]\[K_{\text{sp}} = [Ag^+]^3[PO_4^{3-}]\]Silver phosphate's relatively low \(K_{\text{sp}}\) limits its solubility, which has implications in processes where low concentrations of silver ions are necessary.
In the context of electrochemistry, when used in voltaic cells, as shown in the exercise, the concentration of silver ions from the saturated solution of silver phosphate is a determinant factor in calculating the potential of the cell using the Nernst equation.
This establishes a relationship between the solubility of the compound and its electrolytic behavior, highlighting the interconnectedness in chemical principles.