Question

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

Short answer

In summary, cell potentials are not multiplied by the coefficients in the balanced redox equation because they are an intensive property and remain constant regardless of the reaction scale. Only the Gibbs Free Energy ($\mathrm{\Delta }G$) is affected by the change in coefficients, as demonstrated by the relation $\mathrm{\Delta }{G}_2=k\mathrm{\Delta }{G}_1$ when the balanced redox equation is multiplied by a factor $k$. The relationship between $\mathrm{\Delta }G$ and cell potential ($E_{cell}$) is given by $\mathrm{\Delta }G=-nF{E}_{cell}$, where $n$ is the number of moles of electrons transferred, but this relationship does not require the multiplication of cell potential by the coefficients.

Step-by-step solution

Define Gibbs Free Energy and Cell Potential

Gibbs Free Energy $(\mathrm{\Delta }G)$ is a thermodynamic quantity that represents the energy available in a system to perform work. Cell potential $(E_{cell})$, on the other hand, is a measure of the voltage or electrical potential difference between two half-cells in an electrochemical cell. For a redox reaction in an electrochemical cell, the Gibbs Free Energy change is given by: $$\mathrm{\Delta }G=-nF{E}_{cell}$$ where: $\mathrm{\Delta }G$ = change in Gibbs Free Energy (J/mol) $n$ = number of moles of electrons transferred in the redox reaction $F$ = Faraday's constant, the charge per mole of electrons, approximately 96485 C/mol $E_{cell}$ = cell potential (V)

Analyze how multiplying reaction with coefficients affects $\mathrm{\Delta }G$ and $E_{cell}$

Let's consider a balanced redox reaction with some coefficients and write the expression for both $\mathrm{\Delta }G$ and $E_{cell}$. $$aA+bB\to cC+dD$$ Suppose we multiply this equation by some factor $k$. The new balanced equation is: $$kaA+kbB\to kcC+kdD$$ Now we will analyze the effect on $\mathrm{\Delta }G$ and $E_{cell}$. For the original reaction: $$\mathrm{\Delta }{G}_1=-n_{1}F{E}_{cel{l}_1}$$ For the new (scaled) reaction: $$\mathrm{\Delta }{G}_2=-n_{2}F{E}_{cel{l}_2}$$ Since the number of moles of electrons transferred has also been multiplied by $k$, we have: $n_2=k{n}_1$ And because the cell potential is an intensive property that remains the same even if the reaction is scaled (unaffected by multiplying coefficients): $E_{cel{l}_2}=E_{cel{l}_1}$ Therefore, $$\mathrm{\Delta }{G}_2=-k{n}_{1}F{E}_{cel{l}_1}=k\mathrm{\Delta }{G}_1$$

Explain why coefficients do not affect $E_{cell}$

As per the previous step, we demonstrated that the relationship between $\mathrm{\Delta }{G}_1$ and $\mathrm{\Delta }{G}_2$ is: $$\mathrm{\Delta }{G}_2=k\mathrm{\Delta }{G}_1$$ This shows that the change in Gibbs Free Energy is scaled by a factor of $k$ when the reaction is multiplied by coefficients. However, cell potential $E_{cell}$ remains the same, as it is an intensive property that is independent of scale. In summary, cell potentials are not multiplied by the coefficients in the balanced redox equation because it is an intensive property, and only the Gibbs Free Energy is affected by the change in coefficients.