A silver concentration cell is constructed with the electrolyte at both electrodes being initially \(0.10 \mathrm{M} \mathrm{AgNO}(\mathrm{aq})\) at \(25^{\circ} \mathrm{C}\). The electrolyte at one electrode is diluted by a factor of 10 five times and the cell potential measured each time. (a) Plot the potential of this cell on a graph as a function of \(\ln \left[\mathrm{Ag}^{+}\right]_{\text {anode- }}\) (b) Calculate the value of the slope of the line. To what term in the Nernst equation does this value correspond? Is the value you determined from the plot consistent with the value you would calculate from the values in that term? If not consistent, calculate your percentage error. (c) What is the value of the \(y\)-intercept? To what term in the Nernst equation does this value correspond?

Step by step solution

01

- Understand the Concentration Cell

A concentration cell is a type of electrochemical cell where both electrodes are made of the same metal, but they are immersed in solutions of the metal's ions that have different concentrations. The cell generates a potential due to the concentration difference according to the Nernst equation. For the given silver concentration cell, the electrolyte concentration at one electrode is diluted by a factor of 10 five times, which changes the concentration and thus the cell potential each time.

02

- Implement Experimental Procedure

To create the plot, dilute the silver ion concentration at the anode by a factor of 10 and measure the cell potential each time. Do this process five times, resulting in six concentrations in total: 0.10 M, 0.010 M, 0.0010 M, 0.00010 M, 0.000010 M, and 0.0000010 M. The potential measurements are plotted against the natural logarithm of these concentrations.

03

- Calculate the Slope of the Line

Once the potential values are plotted against \(\ln[\text{Ag}^+]_{\text{anode}}\), a linear relationship should be visible. Calculate the slope of the line using the equation \(slope = (y_2 - y_1) / (x_2 - x_1)\), where \((x_1,y_1)\) and \((x_2,y_2)\) are two points on the line. According to the Nernst equation, this slope should correspond to the term \(-\frac{RT}{nF}\).

04

- Compare the Calculated Slope with Theoretical Value

Compare the slope obtained from the graph to the theoretical value of \(-\frac{RT}{nF}\), where \(R\) is the universal gas constant, \(T\) is the temperature in Kelvin, \(n\) is the number of moles of electrons transferred in the cell reaction, and \(F\) is Faraday's constant. The theoretical value can be calculated using the given temperature of \(25^\circ C\) or \(298 K\). If the values are not consistent, calculate the percentage error.

05

- Determine the Y-intercept and Its Significance

The y-intercept of the line can be determined from the graph. This y-intercept should correspond to the term \(\frac{E^\circ}{0.05916 V}\) in the Nernst equation where \(E^\circ\) is the standard electrode potential. Note the units of the y-intercept which should be in volts.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Nernst Equation

The Nernst equation is a fundamental equation in electrochemistry that relates the voltage or potential of an electrochemical cell to the concentration of ions. It's essential for predicting the potential of an electrolytic cell under non-standard conditions.

The equation is written as: \[ E = E^\circ - \frac{RT}{nF}\ln\left(\frac{[\text{Red}]}{[\text{Ox}]}

\(E\) is the cell potential under non-standard conditions,

\(E^\circ\) is the standard electrode potential,

\(R\) is the universal gas constant,

\(T\) is the temperature in Kelvin,

\(n\) is the number of moles of electrons transferred in the cell reaction,

\(F\) is Faraday's constant,

\([\text{Red}]\) and \([\text{Ox}]\) are the concentrations of the reduced and oxidized species respectively.

In a concentration cell, the equation simplifies as the standard potentials cancel out, since both electrodes are identical. This leaves the cell potential dependent only on the ion concentrations.

Electrochemical Cell Potential

Electrochemical cell potential, often referred to as cell voltage or electromotive force (emf), is the measure of the voltage output of an electrochemical cell. It signifies the ability of the cell to drive an electrical current through an external circuit. In the context of a concentration cell, the cell potential is generated by the difference in concentration of ions at the two electrodes.

For a concentration cell, this can be expressed by a modified Nernst equation where the cell potential changes as the ion concentration changes. In the exercise given, the silver concentration cell generates different potentials as the silver ion concentration at one electrode is systematically diluted, affecting the overall potential of the cell.

Silver Ion Concentration

Silver ion concentration refers to the molarity of silver ions in solution, which is a critical factor in determining the behavior of silver-based electrochemical cells. In a concentration cell, this concentration difference is what drives the electrochemical reaction, thus generating a potential.

In the exercise, diluting the silver ion concentration changes the potential measured at the cell. By plotting these changes against the natural logarithm of the silver ion concentration, the graph illustrates the effect concentration has on the cell potential. The relationship is expected to be linear according to the Nernst equation.

Standard Electrode Potential

The standard electrode potential, designated as \(E^\circ\), is a measure of the intrinsic tendency of a reversible electrode to lose or gain electrons when under standard conditions. It is measured in volts and provides a reference for the measurement of the potential of other electrodes. In the exercise, the standard electrode potential is used along with the Nernst equation to interpret the y-intercept of the plotted graph.

The y-intercept from the graph represents the point where the natural logarithm of the ion concentration is zero. At this point, the cell potential should be equal to the standard electrode potential. This is significant as it allows us to confirm if the experimental conditions align with the theoretical predictions of the Nernst equation.