Question

What is the \(\mathrm{Zn}^{2+}: \mathrm{Cu}^{2+}\) concentration ratio in the following cell at \(25^{\circ} \mathrm{C}\) if the measured cell potential is \(1.07 \mathrm{~V} ?\) $$ \mathrm{Zn}(s)\left|\mathrm{Zn}^{2+}(a q) \| \mathrm{Cu}^{2+}(a q)\right| \mathrm{Cu}(s) $$

Short answer

The concentration ratio \( \mathrm{Zn}^{2+} : \mathrm{Cu}^{2+} \) is approximately 10.23.

Step-by-step solution

Identify the Half-Reactions

First, identify the oxidation and reduction half-reactions for the electrochemical cell. The oxidation half-reaction involves zinc losing electrons: \( \mathrm{Zn}(s) \rightarrow \mathrm{Zn}^{2+}(aq) + 2e^- \). The reduction half-reaction involves copper ions gaining electrons: \( \mathrm{Cu}^{2+}(aq) + 2e^- \rightarrow \mathrm{Cu}(s) \).

Write the Overall Cell Reaction

Combine the half-reactions to write the overall cell reaction: \( \mathrm{Zn}(s) + \mathrm{Cu}^{2+}(aq) \rightarrow \mathrm{Zn}^{2+}(aq) + \mathrm{Cu}(s) \). Here, zinc is oxidized and copper ions are reduced.

Identify Standard Cell Potentials

Using standard reduction potential tables, find the standard reduction potential for each half-reaction. The standard reduction potential for \( \mathrm{Cu}^{2+}/\mathrm{Cu} \) is \( +0.34 \mathrm{~V} \) and for \( \mathrm{Zn}^{2+}/\mathrm{Zn} \) is \( -0.76 \mathrm{~V} \).

Calculate the Standard Cell Potential

Calculate the standard cell potential \( E^\circ_{cell} \) as: \[ E^\circ_{cell} = E^\circ_{\mathrm{reduction}} - E^\circ_{\mathrm{oxidation}} = 0.34 \mathrm{~V} - (-0.76 \mathrm{~V}) = 1.10 \mathrm{~V}. \]

Apply the Nernst Equation

The Nernst equation is given by \[ E_{cell} = E^\circ_{cell} - \frac{RT}{nF} \ln{Q} \]Where \( n = 2 \), \( R = 8.314 \mathrm{~J~mol^{-1}~K^{-1}} \), \( F = 96485 \mathrm{~C~mol^{-1}} \), and \( T = 298 \mathrm{~K} \). Simplifying for room temperature:\[ E_{cell} = E^\circ_{cell} - \frac{0.0592}{n} \log{Q}. \]Substitute \( E_{cell} = 1.07 \mathrm{~V} \) and solve for \( Q \):\[ 1.07 \mathrm{~V} = 1.10 \mathrm{~V} - \frac{0.0592}{2} \log{Q}. \]

Solve for Reaction Quotient \( Q \)

Rearrange the equation to solve for \( \log{Q} \):\[ \log{Q} = \frac{(1.10 - 1.07) \times 2}{0.0592} \approx 1.01 \].Hence, \( Q = 10^{1.01} \approx 10.23 \).

Calculate the Concentration Ratio

The reaction quotient \( Q \) is equivalent to the concentration ratio of reactants divided by products:\[ Q = \frac{[\mathrm{Zn}^{2+}]}{[\mathrm{Cu}^{2+}]} \].Thus, the concentration ratio \( \frac{[\mathrm{Zn}^{2+}]}{[\mathrm{Cu}^{2+}]} \approx 10.23 \).

Key concepts

Electrochemical Cell

Electrochemical cells are fascinating devices that convert chemical energy into electrical energy. These cells work based on redox reactions, which involve the transfer of electrons between substances. In a typical electrochemical cell, such as one involving zinc and copper, there are two main compartments, each housing an electrode.
- **Oxidation occurs at the anode:** Here, zinc metal ( Zn(s) ) undergoes oxidation, losing electrons to form zinc ions ( Zn^{2+}(aq) ). - **Reduction happens at the cathode:** The copper ions ( Cu^{2+}(aq) ) gain electrons to become solid copper ( Cu(s) ).
The electron flow generated between these two electrodes can be harnessed to perform work, such as powering a device. Electrochemical cells can be categorized as galvanic cells (spontaneous reactions) or electrolytic cells (non-spontaneous reactions). Galvanic cells, like the zinc-copper cell, operate spontaneously, generating an electric current as they proceed.
Cell potential, measured in volts, indicates the driving force behind the electron movement. A positive cell potential means the reaction is spontaneous, while a negative one implies it isn't.

Standard Reduction Potential

The concept of standard reduction potential ( E^ ) is fundamental in understanding how electrochemical cells function. It reflects the tendency of a chemical species to be reduced, measured under standard conditions: 1 M concentrations, 25°C (298 K), and 1 atm pressure.
Using these values, you can predict cell reactions and voltages used to build batteries and fuel cells.
- **Positive standard potential:** Indicates a strong tendency to gain electrons and undergo reduction, as seen with copper ( Cu^{2+}/Cu ) at +0.34 V. - **Negative standard potential:** Suggests a lesser tendency for reduction, which is the case for zinc ( Zn^{2+}/Zn ) at -0.76 V.
When calculating the potential of an electrochemical cell, subtract the anode's standard potential from that of the cathode. The higher the absolute value, the stronger the tendency for the reaction to occur. By understanding which species are more likely to gain or lose electrons, we can determine both the cell's voltage and its capacity to do work.

Reaction Quotient

In electrochemical systems, the reaction quotient (Q) is a valuable measurement. It helps to understand how an electrochemical cell's potential deviates from the standard conditions.
The reaction quotient is calculated much like the equilibrium constant (K), using the concentrations ([]) of the reactants and products:
\[Q = \frac{[\text{products}]}{[\text{reactants}]}\]
In the case of the zinc-copper cell:
- Calculate it as the ratio of the concentration of zinc ions (Zn^{2+}]) to copper ions ([Cu^{2+}]).
Through the Nernst Equation, we relate the cell potential E_{cell} when it isn't at standard conditions, to Q. This lets us find unknown concentrations in a solution when the measured potential deviates from the calculated theoretical one. If E_{cell} is measured and E^ is known, the Nernst Equation allows for finding Q, and thereby the concentrations of reactants and products at that moment in the reaction.