Question

Explain why cell potentials are not multiplied by the coefficients in the balanced redox equation. (Use the relationship between \(\Delta G\) and cell potential to do this.)

Short answer

Cell potentials are not multiplied by the coefficients in the balanced redox equation because the Gibbs free energy change (∆G) per one redox event remains the same, even if the coefficients are multiplied by a constant. The cell potential (E) is directly proportional to ∆G and inversely proportional to the number of moles of electrons transferred (n), as shown by the expression \(E = -\frac{\Delta G}{nF}\). Since ∆G remains constant and n increases proportionally with the coefficients, their quotient (E) does not change, demonstrating that cell potentials remain unaffected by the coefficients in the balanced redox equation.

Step-by-step solution

Recognize the relationship between Gibbs free energy and cell potential

The relationship between Gibbs free energy change (∆G) and cell potential (E) is given by the following formula: \[\Delta G = -nFE\] Here, ∆G represents Gibbs free energy change, n is the number of moles of electrons transferred in the redox reaction, F is Faraday's constant, and E is the cell potential.

Derive the expression for cell potential in terms of Gibbs free energy

To find an expression for E in terms of ∆G, we can rearrange the formula in Step 1: \[E = -\frac{\Delta G}{nF}\] Now, E is expressed in terms of ∆G, n, and F.

Discuss the effect of coefficients on Gibbs free energy change

In a balanced redox equation, the coefficients represent the stoichiometry of the reaction, i.e., the ratios in which reactants combine to form products. When we multiply all coefficients of a redox equation by a constant, it signifies the same overall redox process that happened multiple times. The Gibbs free energy change (∆G) for a reaction depends on the change in the mole amounts of products and reactants, driven by the chemical reaction. For one complete redox event, ∆G remains the same, even if the coefficients are multiplied by a constant. This is because the change in energy state from reactants to products remains the same per redox event.

Explain why cell potentials are not multiplied by coefficients

From the expression derived in Step 2, we see that cell potential (E) is directly proportional to the Gibbs free energy change (∆G) and inversely proportional to the number of moles of electrons transferred (n). Since multiplying the coefficients in a balanced redox equation by any constant does not change the overall Gibbs free energy change (∆G) per one redox event, and n also increases proportionally (as mentioned in Step 3), their quotient remains the same. Therefore, the cell potential (E) remains unaffected, which illustrates that cell potentials are not multiplied by the coefficients in the balanced redox equation.

Key concepts

Gibbs Free Energy

When exploring the realm of electrochemistry, one cannot disregard the significance of Gibbs free energy, denoted as \(\Delta G\). It's a vital thermodynamic quantity that indicates whether a process will occur spontaneously at constant temperature and pressure. In simpler terms, \(\Delta G\) measures the amount of 'useful' energy that can be harnessed from a chemical reaction.

A negative \(\Delta G\) suggests a reaction is spontaneous, meaning it can occur without any added energy. On the other hand, a positive value means energy needs to be supplied for the reaction to proceed.

In the context of cell potentials, it's crucial to understand that the Gibbs free energy change relates directly to the electrical work that can be extracted from a redox reaction. The formula \(\Delta G = -nFE\) bridges the gap between the chemical and electrical realms: \(n\) represents the number of moles of electrons exchanged, \(F\) is the Faraday constant (a specific quantity of electric charge per mole of electrons), and \(E\) signifies the cell potential or the electromotive force (EMF) of the reaction.

When the coefficients of a balanced redox equation are multiplied by a constant, it's akin to running the same 'electrical program' several times; the energy per 'run' doesn't change, much like how copying a file on a computer doesn't make the file larger. Thus, despite stoichiometry adjustments, \(\Delta G\) for a redox process remains consistent per electron transfer, ensuring the cell potential—a measure of energy per charge—is constant.

Redox Reaction

Redox reactions are the heart of electrochemical processes, involving the transfer of electrons between two substances. Redox stands for 'reduction-oxidation', with reduction pertaining to the gain of electrons and oxidation to the loss. It's a synchronized dance where one reactant gives up electrons while another gladly accepts them.

Each participant in a redox reaction plays a defined role: the oxidizing agent gets reduced while the reducing agent is oxidized. The coefficients in these reactions reflect the precise balance required for the electron exchange to occur without any leftovers on either side.

Understanding that cell potential is hinged on the energy per charge, not the total number of particles, is pivotal. Just as a singer's volume isn't compounded by singing the same note repeatedly, the intensity of a redox reaction, measured by its cell potential, isn't compounded by repeating the electron transfer.

A balanced redox equation with altered coefficients still encodes the same reaction, just on a different scale, without modifying the energy cost or gain of a single electron transfer, which is ultimately, the essence of the cell potential.

Electrochemistry

Electrochemistry is the study of chemical reactions that produce electrical currents and the use of electricity to drive chemical transformations. It's a meeting ground for electrons, chemicals, and energy, with implications strewn from batteries to biosensors.

Central to electrochemistry is the concept of cell potential, which is essentially a voltage that reflects the ability of a redox reaction to deliver energy. It signifies how forcefully electrons are pushed or pulled through a circuit—a higher cell potential means a greater capability to do work, akin to the higher pressure in a water hose producing a stronger stream.

The meticulous composition of the electrochemical cell, the chemistries at play, and the electrodes' material all define the cell's potential. However, the size of the reaction (dictated by its coefficients) does not impact this potential, much like how the force of gravity on Earth doesn't increase just because more people are standing on the ground. Thus, electrochemistry teaches us that while the quantity of reactants and products may vary, the inherent energy per unit charge remains untouched, maintaining a constant cell potential.