Animated Solution for Chemistry - Electrochemistry: The cell, $\text{Zn}|{\text{Zn}}^{2+}(1\text{M})||{\text{Cu}}^{2+}(1\text{M})|\text{Cu}$ ($E_{\text{cell}}^{\circ }=1.10\text{ V}$), was allowed to be completely discharged at 298 K. The relative concentration of ${\text{Zn}}^{2+}$ to ${\text{Cu}}^{2+}$ $\left(\frac{[{\text{Zn}}^{2+}]}{[{\text{Cu}}^{2+}]}\right)$ is

Imagine you are using a battery to power a flashlight. Over time, the light dims and eventually goes out. We say the battery is "dead" or completely discharged. But what is actually happening inside the battery at a chemical level?

When an electrochemical cell is completely discharged, it means the chemical reaction driving the electron flow has reached equilibrium. At this point, there is no longer a net driving force to push electrons through the external circuit. Mathematically, this means the cell potential, $E_{\text{cell}}$, has dropped exactly to zero.

Nernst

To understand the relationship between the cell potential and the concentrations of the ions, we rely on the mighty Nernst Equation:

Here, $E_{\text{cell}}^{\circ }$ is the standard cell potential, $n$ is the number of moles of electrons transferred in the balanced redox reaction, and $Q$ is the reaction quotient.

For our specific Daniel cell, the overall reaction is:

Notice that solid Zinc and solid Copper do not appear in our reaction quotient because their concentrations are constant. Therefore, $Q=\frac{[{\text{Zn}}^{2+}]}{[{\text{Cu}}^{2+}]}$. Also, since Zinc loses 2 electrons and Copper gains 2 electrons, we know that $n=2$.

As we established earlier, a discharged cell is at equilibrium. At equilibrium, two critical things happen: 1. $E_{\text{cell}}=0$ 2. $Q=K_{\text{eq}}$

Let's substitute these conditions into our Nernst equation:

Now, it is just a matter of careful algebraic rearrangement. We want to isolate $\log K_{\text{eq}}$:

To find the actual ratio of the ions, which is our equilibrium constant $K_{\text{eq}}$, we take the antilog of both sides:

This massive number tells us that at equilibrium, the concentration of Zinc ions is astronomically higher than the concentration of Copper ions. The reaction heavily favors the products, which aligns perfectly with the positive standard cell potential!