Question

Given \(\mathrm{Zn} \rightarrow \mathrm{Zn}^{+2}+2 \mathrm{e}^{-}\) with \(\mathrm{E}^{\circ}=+.763\), calculate \(\mathrm{E}\) for a Zn electrode in which \(\mathrm{Zn}^{+2}=.025 \mathrm{M}\)

Short answer

The actual cell potential (E) for the Zn electrode in a Zn²⁺ solution with a concentration of 0.025 M is approximately 0.803 V.

Step-by-step solution

Understand the Nernst equation

The Nernst equation enables us to calculate the cell potential (E) based on the standard cell potential (E°), concentrations, and the number of electrons exchanged in the redox reaction. The Nernst equation is given as: \[E = E° - \frac{RT}{nF} \ln Q\] Where: - E is the cell potential to be calculated - E° is the standard cell potential (+0.763 V) - R is the gas constant (8.314 J/mol·K) - T is the temperature in Kelvin (assume 298 K for room temperature) - n is the number of electrons transferred in the redox reaction (2 for Zn) - F is the Faraday constant (96485 C/mol) - Q is the reaction quotient, calculated using the given concentrations (0.025 M for Zn²⁺)

Calculate the reaction quotient (Q)

As the given reaction is a reduction half-reaction, the reaction quotient (Q) can be calculated only using the concentration of the zinc ion (Zn²⁺). The balanced reaction is: \[\mathrm{Zn^{2+}} + 2\mathrm{e^-} \rightarrow \mathrm{Zn}\] Hence, Q becomes: \[Q = [\mathrm{Zn}^{2+}]\] From the exercise, the given concentration of Zn²⁺ is 0.025 M: \[Q = 0.025\]

Apply the Nernst equation

With the computed value of Q, all parameters are now available to be plugged into the Nernst equation: \[E = E° - \frac{RT}{nF} \ln Q\] \[E = 0.763 - \frac{(8.314)(298)}{(2)(96485)} \ln 0.025\] Now, you can calculate the value of E: \[E \approx 0.803\, V\]

State the final answer

The actual cell potential (E) for the Zn electrode in a Zn²⁺ solution with a concentration of 0.025 M is approximately 0.803 V.

Key concepts

Understanding Cell Potential

The concept of cell potential is crucial in electrochemistry, as it determines the electrical energy a cell can produce. The cell potential, often denoted as \( E \), can be considered as a measure of the driving force behind an electrochemical reaction. When a redox reaction takes place, electrons transfer from one species to another. This electron flow, which occurs due to potential difference, can do work—like powering a device. The standard cell potential, \( E^\circ \), measures this potential when each reactant and product is in its standard state. It provides a reference point, typically at concentrations of 1 M, pressure of 1 atm, and temperature of 298 K (25°C). However, under non-standard conditions, you need tools like the Nernst equation to find the cell potential.

The Mechanism of Redox Reactions

Redox reaction, short for reduction-oxidation reaction, involves the transfer of electrons between two species. One species undergoes oxidation by losing electrons, while the other is reduced by gaining those electrons. It is essential to identify the two half-reactions in any redox process—namely, oxidation and reduction.
For instance, zinc (Zn) in the reaction \( \mathrm{Zn} \rightarrow \mathrm{Zn^{2+}} + 2 \mathrm{e^-} \) undergoes oxidation as it loses electrons. Meanwhile, in return, it can also gain electrons in a reduction process involving zinc ions: \( \mathrm{Zn^{2+}} + 2 \mathrm{e^-} \rightarrow \mathrm{Zn} \). This electron exchange is what contributes to the electrochemical cell potential.

The Role of Reaction Quotient in Calculating Potential

The reaction quotient, \( Q \), is vital for understanding how a reaction proceeds towards equilibrium under non-standard conditions. It reflects the ratio of the concentrations of reactants to products at any point in the reaction. Calculating \( Q \) is essential for applying the Nernst equation effectively, as it allows us to adjust the standard cell potential to the actual conditions of the reaction.
In the given exercise, \( Q \) was calculated using only the concentration of zinc ions, yielding \( Q = 0.025 \). This was plugged into the Nernst equation for adjustments due to the deviation from standard conditions. With \( Q \) and other constants, you can find the actual cell potential, reflecting real-world scenarios where concentrations rarely remain at ideal states.