Question

At 298 \(\mathrm{K}\) a cell reaction has a standard cell potential of \(+0.17 \mathrm{V} .\) The equilibrium constant for the reaction is \(5.5 \times 10^{5} .\) What is the value of \(n\) for the reaction?

Short answer

The value of $n$ for the reaction is approximately 2, which represents the number of electrons transferred in the reaction. This is calculated using the Nernst Equation, given the standard cell potential, the temperature, and the equilibrium constant.

Step-by-step solution

Write down the Nernst Equation

The Nernst Equation relates the standard cell potential (E°), the number of electrons transferred (n), the equilibrium constant (K), and temperature (T): \[ E° = \frac{RT}{nF} \ln{K} \] where R is the gas constant (8.314 J/(mol·K)), T is the temperature in Kelvin, and F is the Faraday's constant (96485 C/mol).

Plug in the given values and isolate n

We have E° = 0.17 V, T = 298 K, and K = 5.5 × 10^5. Plugging these values into the equation, we get: \[ 0.17 = \frac{(8.314 \,\text{J}\, (\text{mol} \cdot \text{K})^{-1})(298 \,\text{K})}{n(96485 \,\text{C/mol})} \ln(5.5 \times 10^{5}) \] Now, rearrange the equation to isolate n: \[ n = \frac{(8.314 \,\text{J}\, (\text{mol} \cdot \text{K})^{-1})(298 \,\text{K})}{0.17 \,\text{V}} \cdot \frac{1}{(96485 \,\text{C/mol})} \cdot \frac{1}{\ln(5.5 \times 10^{5})} \]

Calculate the value of n

Now, calculate the value of n: \[ n = \frac{(8.314)(298)}{0.17} \cdot \frac{1}{(96485)} \cdot \frac{1}{\ln(5.5 \times 10^{5})} \] \[ n \approx 1.956 \]

Round the value of n to the nearest integer

Since n represents the number of electrons transferred in the reaction, it must be a whole number. In this case, we can round n to the nearest integer: \[ n \approx 2 \] So, the value of n for the reaction is approximately 2.

Key concepts

Standard Cell Potential

In the realm of electrochemistry, the term standard cell potential, often represented as E°, is a critical component in assessing the ability of an electrochemical cell to do work when no current is being drawn. It is a measure of the electromotive force, or emf, of a cell when all reactants and products are at their standard states, typically at a concentration of 1 molar and a pressure of 1 atmosphere.

It is expressed in volts (V) and serves as a key indicator of whether a redox reaction will occur spontaneously. A positive value of E° suggests that the reaction can occur spontaneously under standard conditions, whereas a negative value indicates a non-spontaneous reaction. For example, in the given exercise, the standard cell potential for the cell reaction is +0.17 V, which implies that the reaction tends toward occurring spontaneously.

Incorporating the standard cell potential into the Nernst Equation allows for the calculation of the equilibrium constant, and in turn, provides the number of electrons transferred during the reaction, illustrated as n in the exercise. Understanding this concept is paramount for students tackling electrochemistry problems, as it directly relates to the spontaneity and efficiency of electrochemical reactions.

Equilibrium Constant

The concept of the equilibrium constant, designated as K, plays a central role in understanding chemical reactions at equilibrium. In electrochemistry, equilibrium state implies a balance where the forward and reverse reactions occur at the same rate, resulting in no net change in the concentration of reactants and products over time.

The equilibrium constant quantifies the ratio of product concentrations to reactant concentrations, with each raised to the power of their respective stoichiometric coefficients. In the context of the Nernst Equation, a larger value of K, suggests a reaction heavily favors the formation of products under standard conditions, which is key for understanding the direction and extent of the reaction.

For instance, the exercise presents an equilibrium constant K of 5.5 × 10^5, indicating a significant preference for product formation. The Nernst Equation elegantly ties together the equilibrium constant and the standard cell potential, offering insights into the potential and directionality of redox reactions. Such knowledge not only aids in solving homework problems but also provides a foundation for predicting the behavior of chemical systems.

Electrochemical Cell

An electrochemical cell is a device capable of either generating electrical energy from chemical reactions or facilitating chemical reactions through the introduction of electrical energy. These cells are the building blocks of batteries, fuel cells, and electrolysis systems. The two types of electrochemical cells are galvanic (or voltaic) cells, which convert chemical energy into electrical energy, and electrolytic cells, which do the opposite.

In a galvanic cell, a spontaneous redox reaction drives the flow of electrons from the anode to the cathode, creating an electric current. This is the type of cell typically implied in standard cell potential discussions, where the cell's ability to do electrical work stems from a natural tendency for reaction.

Understanding the components and functions of electrochemical cells is crucial for comprehending the broader implications of the Nernst Equation. This equation provides insight into how variables such as temperature, pressure, and concentration affect the cell's potential and behavior. Through practical exercises, such as the one with the given standard cell potential and equilibrium constant, students can visualize the inner workings of these cells, enhancing their grasp on the dynamic relationships within electrochemical systems.