Problem 57 - Chemistry The Central Science

Question

A cell has a standard cell potential of \(+0.177 \mathrm{~V}\) at \(298 \mathrm{~K}\). What is the value of the equilibrium constant for the reaction (a) if \(n=1 ?(\mathbf{b})\) if \(n=2 ?(\mathbf{c})\) if \(n=3 ?\)

Step-by-Step Solution

Answer

For a cell with a standard cell potential of \(+0.177 V\) at \(298 K\), the equilibrium constants for different values of the number of electrons transferred (n) are: a) n=1: \(K \approx 1871\) b) n=2: \(K \approx 3.494\times10^6\) c) n=3: \(K \approx 6.532\times10^{9}\)

Step 1: Rearrange the Nernst Equation

We need to isolate K in the Nernst equation, so first rearrange it as: \( K = e^\frac{nFE_\text{cell}}{RT} \) Now we can plug in the given values and constants to find K for each case.

Step 2: Case a (n=1)

Insert the given values and constants with n=1: \( K = e^\frac{(1)(96485 C/mol)(0.177 V)}{(8.314 J/mol K)(298 K)} \) Calculate the equilibrium constant (K) for n=1: \(K = e^{7.529}\) \(K \approx 1871\)

Step 3: Case b (n=2)

Insert the given values and constants with n=2: \( K = e^\frac{(2)(96485 C/mol)(0.177 V)}{(8.314 J/mol K)(298 K)} \) Calculate the equilibrium constant (K) for n=2: \(K = e^{15.058}\) \(K \approx 3.494\times10^6 \)

Step 4: Case c (n=3)

Insert the given values and constants with n=3: \( K = e^\frac{(3)(96485 C/mol)(0.177 V)}{(8.314 J/mol K)(298 K)} \) Calculate the equilibrium constant (K) for n=3: \(K = e^{22.587}\) \(K \approx 6.532\times10^{9}\) In summary, the equilibrium constants for the different values of n are: a) n=1: \(K \approx 1871\) b) n=2: \(K \approx 3.494\times10^6\) c) n=3: \(K \approx 6.532\times10^{9}\)

Key Concepts

Nernst Equation

The Nernst equation is a fundamental formula in electrochemistry that relates the reduction potential of a chemical reaction to the concentration of the reactants and products. Understanding the Nernst equation is crucial when dealing with electrochemical cells and predicting the direction in which a reaction will proceed.

At its core, the equation gives us a way to calculate the potential of an electrochemical cell under non-standard conditions. It is expressed as:
\[ E = E^\circ - \frac{RT}{nF} \ln(Q) \]
Where:

• \(E\) is the cell potential at non-standard conditions,
• \(E^\circ\) is the standard cell potential,
• \(R\) is the universal gas constant,
• \(T\) is the temperature in Kelvin,
• \(n\) is the number of moles of electrons exchanged,
• \(F\) is the Faraday's constant,
• \(Q\) is the reaction quotient.

The equation becomes particularly simple when conditions are at equilibrium, as the cell potential (\(E\)) becomes zero, and \(Q\) is replaced with the equilibrium constant (\(K\)). Thus, the equation is modified to calculate \(K\), relating the standard cell potential directly to the equilibrium constant of the reaction.

Standard Cell Potential

Standard cell potential (\(E^\circ\)) is a measure of how forcefully an electrochemical cell can push electrons through a circuit under standard conditions. These standard conditions usually involve all solutes being at 1 M concentration and gases at 1 bar pressure, with the temperature typically at 298 K (25°C).

The value of \(E^\circ\) is significant in determining the natural tendency for a reaction to occur. A positive standard cell potential indicates a spontaneous reaction under standard conditions, while a negative value implies a non-spontaneous reaction.

In the given exercise, the standard cell potential is marked as positive (+0.177 V), making it a crucial factor in predicting the spontaneity of the reaction and, therefore, its equilibrium constant, as we can see in the Nernst equation.

Chemical Equilibrium

Chemical equilibrium represents the state in a chemical reaction where the rates of the forward and reverse reactions are equal, resulting in no net change in the concentration of the reactants and products. The position of the equilibrium, reflected in the equilibrium constant (\(K\)), tells us the extent to which a reaction will occur before reaching this state.

The equilibrium constant is calculated from the concentrations of the reactants and products at equilibrium. For a generic reaction \(aA + bB ⇌ cC + dD\), the equilibrium constant expression is written as:
\[ K = \frac{[C]^c[D]^d}{[A]^a[B]^b} \]
Where the square brackets denote the concentration of each species. In the context of electrochemistry, when considering a redox reaction, the standard cell potential gives us insight into the equilibrium constant without needing the concentration of individual species. The higher the value of \(K\), the more products are favored, as seen above where \(K\) increases with the number of moles of electrons exchanged.

Electrochemistry

Electrochemistry is a branch of chemistry that deals with the interrelation of electrical currents and chemical reactions. Two primary devices used in this field are electrochemical cells, which convert chemical energy into electrical energy, and electrolytic cells, which do the opposite. Within an electrochemical context, the standard cell potential and the equilibrium constant play pivotal roles in determining the feasibility and extent of chemical reactions.

An understanding of electrochemistry is not complete without a grasp of how the Nernst equation and the standard cell potential relate to the equilibrium of chemical reactions. By using our knowledge of these concepts, we can control and predict the outcomes of a myriad of processes, from the simple batteries that power our everyday devices to the complex reactions that occur in biological systems.