Problem 87 - Chemistry: Principles and Reactions

Question

For the cell $$ \mathrm{Zn}\left|\mathrm{Zn}^{2+}\right| \mathrm{Cu}^{2+} \mid \mathrm{Cu} $$ \(E^{\circ}\) is \(1.10 \mathrm{~V}\). A student prepared the same cell in the lab at standard conditions. Her experimental \(E^{\circ}\) was \(1.0 \mathrm{~V}\). A possible explanation for the difference is that (a) a larger volume of \(\mathrm{Zn}^{2+}\) than \(\mathrm{Cu}^{2+}\) was used. (b) the zinc electrode had twice the mass of the copper electrode. (c) \(\left[\mathrm{Zn}^{2+}\right]\) was smaller than \(1 M\). (d) \(\left[\mathrm{Cu}^{2+}\right]\) was smaller than \(1 M\). (e) the copper electrode had twice the surface area of the zinc electrode.

Step-by-Step Solution

Answer

a) A larger volume of \(\mathrm{Zn}^{2+}\) than \(\mathrm{Cu}^{2+}\): No b) The zinc electrode with twice the mass of the copper electrode: No c) The concentration of \(\mathrm{Zn}^{2+}\) smaller than 1 M: Yes d) The concentration of \(\mathrm{Cu}^{2+}\) smaller than 1 M: Yes e) The copper electrode had twice the surface area of the zinc electrode: No

Step 1: Evaluate option (a)

In this option, a larger volume of \(\mathrm{Zn}^{2+}\) than \(\mathrm{Cu}^{2+}\) was used. Volumes, however, do not affect the cell potential, as the potential difference depends on concentrations and not volumes. So, this option is not a possible explanation for the lower experimental \(E^{\circ}\).

Step 2: Evaluate option (b)

In option (b), we have the zinc electrode with twice the mass of the copper electrode. The overall cell potential is not directly related to the mass of the electrodes. So, this option is also not a possible explanation for the lower experimental \(E^{\circ}\).

Step 3: Evaluate option (c)

Option (c) suggests that the concentration of \(\mathrm{Zn}^{2+}\), \(\left[\mathrm{Zn}^{2+}\right]\), was smaller than 1 M. Since the Nernst equation states that the cell potential difference depends on the concentrations of ions involved in the reaction, a smaller concentration of \(\mathrm{Zn}^{2+}\) could cause the observed difference in \(E^{\circ}\). This option could be a possible explanation.

Step 4: Evaluate option (d)

Option (d) says that the concentration of \(\mathrm{Cu}^{2+}\), \(\left[\mathrm{Cu}^{2+}\right]\), was smaller than 1 M. As discussed for option (c), the cell potential difference depends on the concentrations of ions involved in the reaction. Therefore, a smaller concentration of \(\mathrm{Cu}^{2+}\) could also lead to a difference in \(E^{\circ}\). This option is also a possible explanation.

Step 5: Evaluate option (e)

In option (e), the copper electrode had twice the surface area of the zinc electrode. The surface area does not affect the overall potential difference directly, so this option is not a possible explanation for the lower experimental \(E^{\circ}\).

Step 6: Conclusion

Options (c) and (d) are both possible explanations for the difference in the experimental \(E^{\circ}\). These options state that the concentration of \(\mathrm{Zn}^{2+}\) or \(\mathrm{Cu}^{2+}\) was smaller than 1 M, which could lead to a different cell potential difference observed in the lab experiment.

Key Concepts

Nernst Equation

The Nernst equation is a fundamental concept in electrochemistry, which provides a way to calculate the cell potential, taking into account the effect of ion concentrations. It is especially useful when reactions do not occur under standard conditions. The formula is written as:\[E = E^{°} - \left(\frac{RT}{nF}\right) \ln(Q) \]where:

• \(E\) is the cell potential under non-standard conditions.
• \(E^{°}\) is the standard cell potential.
• \(R\) is the universal gas constant \(8.314 \, \text{J/mol K}\).
• \(T\) is the temperature in Kelvin.
• \(n\) is the number of moles of electrons exchanged.
• \(F\) is the Faraday constant \(96485 \, \text{C/mol}\).
• \(Q\) is the reaction quotient, which represents the ratio of the concentrations of the products over the reactants raised to the power of their stoichiometric coefficients.

The equation sees practical application in situations where the ionic concentrations deviate from 1 M. For instance, in the original exercise, deviations from the standard 1 M concentrations led to changes in cell potential values, revealable via the Nernst equation.

Cell Potential

Cell potential, also known as electromotive force (emf), is the driving force for the movement of electrons in an electrochemical cell from the anode to the cathode. It indicates the voltage of the cell and can be measured in volts (V).

• The cell potential under standard conditions, denoted as \(E^{°}\), is determined when all reactants and products are at 1 M concentration, and the temperature is \(25 \, ^\circ\text{C}\) or \(298 \text{K}\).
• If the cell potential is positive, the reaction is spontaneous, meaning it can occur without external energy input.

In the exercise, the standard cell emf of the zinc-copper cell is \(1.10 \, \text{V}\). However, during the experiment, the observed potential was \(1.0 \, \text{V}\), which implies non-standard conditions. Possible explanations such as variations in ion concentrations were considered, which affect actual cell potential and can be analyzed using the Nernst equation.

Standard Conditions

In electrochemistry, standard conditions refer to the specific set of conditions under which cell potential measurements are made and reported in order to ensure consistency. These conditions include:

• Concentration: All reactants and products should be at 1 M concentration.
• Pressure: Any gases involved should be at a pressure of 1 atmosphere.
• Temperature: The standard temperature is \(25 \, ^\circ\text{C}\) or \(298 \text{K}\).

Under these conditions, the cell potential measured is referred to as the standard cell potential, denoted \(E^{°}\). It's important because it allows chemists to compare different reactions and cells using a common baseline. In experimental labs, deviations from these standard conditions, such as using solutions with less than 1 M concentration, often explain differences in observed cell potentials as opposed to theoretical expectations.