Question

Consider a concentration cell similar to the one shown in Exercise 69, except that both electrodes are made of \(\mathrm{Ni}\) and in the left-hand compartment \(\left[\mathrm{Ni}^{2+}\right]=1.0 M .\) Calculate the cell potential at \(25^{\circ} \mathrm{C}\) when the concentration of \(\mathrm{Ni}^{2+}\) in the compartment on the right has each of the following values. a. \(1.0 M\) b. \(2.0 \mathrm{M}\) c. \(0.10 \mathrm{M}\) d. \(4.0 \times 10^{-5} M\) e. Calculate the potential when both solutions are \(2.5 M\) in \(\mathrm{Ni}^{2+}\). For each case, also identify the cathode, anode, and the direction in which electrons flow.

Short answer

a) Cell potential is \(0 \mathrm{V}\), and there is no electron flow, so no cathode or anode. b) Cell potential is \(0.0148 \mathrm{V}\), left compartment is the cathode, right compartment is the anode, and electrons flow from right to left. c) Cell potential is \(-0.0296 \mathrm{V}\), left compartment is the anode, right compartment is the cathode, and electrons flow from left to right. d) Cell potential is \(-0.0992 \mathrm{V}\), left compartment is the anode, right compartment is the cathode, and electrons flow from left to right. e) Cell potential is \(0 \mathrm{V}\), and there is no electron flow, so no cathode or anode.

Step-by-step solution

Understand the Concentration Cell and Components

A concentration cell is an electrochemical cell that has the same electrode on both sides but different concentrations of the electrolyte on either side. In this problem, we have a \(\mathrm{Ni}\) concentration cell with varying concentrations of \(\mathrm{Ni}^{2+}\) in both compartments. Left compartment: \(\mathrm{Ni^{2+}}\) with concentration \(1.0 M\) Right compartment: \(\mathrm{Ni^{2+}}\) with different concentrations The cell reaction is: \(\mathrm{Ni^{2+}(right) + 2e^- <=> Ni^{2+}(left)}\)

Use the Nernst Equation to Calculate the Cell Potential

The Nernst equation is given by: \(E = E_{0} - \frac{RT}{nF} \ln{\frac{Q}}\) where \(E\) is the cell potential, \(E_{0}\) is the standard cell potential, \(R\) is the gas constant, \(T\) is the temperature, \(n\) is the number of electrons transferred, \(F\) is the Faraday constant, and \(Q\) is the reaction quotient. In this case, the standard potential \(E_{0}\) is \(0 \mathrm{V}\) since both compartments have the same electrode. We will calculate the cell potential for each concentration of \(\mathrm{Ni^{2+}}\) in the right compartment.

Calculate the cell potential for each concentration

Temperature \(T\) is given as \(25^{\circ} \mathrm{C}\) which is equal to \(298.15\) K. The Nernst equation becomes: \(E = - \frac{0.0592 \mathrm{V}}{n} \log{\left(\frac{[\mathrm{Ni^{2+}(left)}]}{[\mathrm{Ni^{2+}(right)]}\right)}\) We will calculate the cell potentials for each case: a) When the concentration of \(\mathrm{Ni^{2+}}\) in the right compartment is \(1.0 M\): \(E = - \frac{0.0592}{2} \log{\left(\frac{1}{1}\right)} = 0 \mathrm{V}\) b) When the concentration of \(\mathrm{Ni^{2+}}\) in the right compartment is \(2.0 M\): \(E = - \frac{0.0592}{2} \log{\left(\frac{1}{2}\right)} = 0.0148 \mathrm{V}\) c) When the concentration of \(\mathrm{Ni^{2+}}\) in the right compartment is \(0.10 M\): \(E = - \frac{0.0592}{2} \log{\left(\frac{1}{0.10}\right)} = -0.0296 \mathrm{V}\) d) When the concentration of \(\mathrm{Ni^{2+}}\) in the right compartment is \(4.0 \times 10^{-5} M\): \(E = - \frac{0.0592}{2} \log{\left(\frac{1}{4.0 \times 10^{-5}}\right)} = -0.0992 \mathrm{V}\) e) When both solutions have concentration \(2.5 M\): \(E = - \frac{0.0592}{2} \log{\left(\frac{2.5}{2.5}\right)} = 0 \mathrm{V}\)

Identify Cathode, Anode, and Electron Flow Direction

For each case: a) The potentials are equal, so there is no electron flow and no cathode or anode. b) The left compartment acts as the cathode, the right compartment acts as the anode, and electrons flow from the right to the left compartment. c) The left compartment acts as the anode, the right compartment acts as the cathode, and electrons flow from the left to the right compartment. d) The left compartment acts as the anode, the right compartment acts as the cathode, and electrons flow from the left to the right compartment. e) The potentials are equal, so there is no electron flow and no cathode or anode.

Key concepts

Concentration cell

A concentration cell is a special type of electrochemical cell. It has two half-cells with the same electrodes and electrolyte type, but at different concentrations. This difference in concentration creates a potential difference, or voltage, between the two electrodes.
In a concentration cell involving nickel ( Ni ), the left and right compartments each contain a nickel electrode. The cell potential arises from the variance in [Ni^{2+}] concentrations. The cell's goal is to reach equilibrium, where both sides have the same concentration.
A fascinating aspect of concentration cells is that they drive the transport of ions from the high-concentration side to the low-concentration side. This establishes a voltage that can be harnessed to perform electrical work, such as powering a small device.

Nernst equation

The Nernst equation is a key tool in electrochemistry, used to calculate the cell potential of an electrochemical cell under non-standard conditions. It is represented as:

\[E = E_{0} - \frac{RT}{nF} \ln{\frac{Q}}\]

where \(E\) is the cell potential, \(E_{0}\) is the standard potential, \(R\) is the gas constant, \(T\) is the temperature in Kelvin, \(n\) is the number of electrons exchanged, \(F\) is the Faraday constant, and \(Q\) is the reaction quotient.

For concentration cells, the standard potential \(E_{0}\) is zero because the half-cells are identical, apart from ion concentrations. The equation simplifies to calculate potential based solely on concentration differences:

\[E = - \frac{0.0592 \text{ V}}{n} \log{\left(\frac{[\text{Concentration left}]}{[\text{Concentration right}]}\right)}\]

At room temperature (25°C), this equation can be used to predict how changing concentrations in the half-cells affects cell potential.

Cell potential

Cell potential, or electromotive force (EMF), is the measure of the voltage between the two electrodes of an electrochemical cell. It indicates the tendency of a cell to drive an electric current through a circuit.
In concentration cells, differences in ionic concentration between the compartments generate a cell potential. This can be calculated using the Nernst equation.

• If the concentration is higher in one compartment than the other, the cell will have a positive or negative potential, depending on which side is more concentrated.
• If concentrations are equal, the cell potential is zero, and no net electron flow occurs.

Understanding cell potential is vital as it helps in determining whether a reaction will be spontaneous and the direction of electron flow. By evaluating cell potential, scientists can determine how to harness and optimize energy from these reactions.

Cathode and anode identification

Identifying the cathode and anode in an electrochemical cell is crucial for understanding electron flow and the overall reaction direction. In a concentration cell:

• The cathode is where reduction occurs. Electrons flow toward the cathode.
• The anode is where oxidation happens. Electrons flow away from the anode.

In scenarios where the left compartment has a higher concentration of Ni^{2+} , it acts as the anode, and the right compartment, being at lower concentration, becomes the cathode. Electrons flow from left to right. Conversely, if the concentration is higher in the right compartment, it becomes the anode and the left is the cathode, with electrons moving from right to left.
In cases where the concentrations are equal, the cell reaches equilibrium, and there is no net electron flow, meaning no distinct cathode or anode.