Cell Potential and Free Energy

Relation Between Gibbs Free Energy and Cell Potential

In animals, nerve cells work by creating a potential difference of positive and negative ions. On one side of the cell there are accumulated Na⁺ ions, and on the other are a lot of K⁺ ions. The separation of these two ions creates a potential difference in the cell. So, when the cell opens channels, and allows this to happen, it produces energy. This is enough energy to stimulate the nerve cell which is what leads to anything you do; whether it is reading this article, sitting down, or even just breathing.

A similar method is used in plant roots, which take up water and nutrients from the soil. A potential difference can be created within the root that allows stuff from the soil to flow into the roots. This concept is also what powers electric cars. In a battery, a potential difference generates power, which is then used to operate the device. This concept of potential difference, when applied to chemical cells, is called cell potential. When electrons flow from a place of high electron density, to a place of low electron density, they generate an electric charge, measured in Volts. This is referred to as a spontaneous reaction and has a negative free energy change.

To better understand the relationship between Gibbs free energy and cell potential, it is necessary that we recap spontaneous and nonspontaneous reactions. Recall that a spontaneous reaction will occur on its own with no extra help, while a nonspontaneous reaction will not occur unless something assists it.

Gibbs Free Energy and Electrochemical Cell Potential

What is the difference between cell potential and electrochemical cell potential? And how do they both relate to Gibbs free energy? Well, the term cell potential is more of an umbrella term that can be used to describe the potential difference in any kind of cell; whether that be biological or artificial. Electrochemical cell potential only describes the potential difference in an artificial cell. Essentially, one that's found in a lab or a battery.

Typically, we just use cell potential to describe everything, but it is important to recognize the slight difference between the two. They both relate to Gibbs free energy in the same way, since they can technically both describe the same thing. Quantifying this relationship is typically done with electrochemical cells, and to accomplish that, we need to know how to calculate cell potential.

Calculating Cell Potential

In the study of Electrolysis, the potential difference between two half-cells is measured with a voltmeter. A positive voltage means that electrons are freely flowing from one cell to another. This means that a positive cell potential corresponds to a spontaneous reaction. For Example:

As the cell works, the zinc electrode will get smaller and the copper cathode will get bigger as electrons move from the anode to the cathode. At the end of the experiment, the copper cathode will be noticeably larger in size.

As electrons move from the Zn anode to the Cu cathode, a number appears on the voltmeter, which is the cell potential, E°. We know that this reaction is spontaneous because it is proceeding without any external help. But we don't need to set up a galvanic cell every time we want to see if a cell potential is positive. It is easier to just calculate the cell potential based on the Standard Potential of each cell using our trusty formula.¹

$$ E^{\circ}_{cell} = E^{\circ}_{cathode} - E^{\circ}_{anode} $$

So, to determine the standard cell potential of the cell, we subtract the standard potential of the anode by the standard potential of the cathode. In this reaction, zinc is giving electrons (oxidation) to copper, which is receiving electrons (reduction).

\begin{align} Cu^{2+} + 2e^- \rightarrow Cu(s) \qquad &E^{\circ} = 0.34~V \\ Zn^{2+} + 2e^- \rightarrow Zn(s) \qquad &E^{\circ} = -0.76~V \end{align}

By combining these two half-cell potentials, we can determine the overall cell potential.

\begin{align} &E^{\circ}_{cell} = E^{\circ}_{cathode} - E^{\circ}_{anode} \\ &E^{\circ}_{cell} = E^{\circ}_{Cu^{2+}/Cu} - E^{\circ}_{Zn^{2+}/Zn} \\ &E^{\circ}_{cell} = 0.34~V - (-0.76~V) \\ &E^{\circ}_{cell} = 1.10~V\end{align}

The overall cell potential of this voltaic cell is +1.10 V, and is thus a spontaneous reaction. But, how do we know which will be the cathode, and which will be the anode? Well, technically, we can pick whatever we want! Try calculating this reaction in the reverse direction—is it still a spontaneous reaction?

For a reaction to occur, there must be a positive cell potential. However, that doesn't mean we can't make a nonspontaneous reaction occur. Electrolysis shows us that if we want to make a nonspontaneous reaction happen, we just apply voltage until the cell potential becomes positive. It's as simple as that! That means that any reaction can be made possible with enough voltage.

Cell Potential vs. Gibbs Free Energy

We know now that cell potential and Gibbs' free energy change both evaluate the spontaneity of a reaction.

So when the cell potential is greater than 0, the reaction will be spontaneous. When Gibbs Free Energy is less than 0, the reaction is spontaneous. That means that cell potential and free energy are inversely proportional.

Derivation of the Relationship Between Free Energy and Cell Potential

It is now pretty simple to see when a reaction will be spontaneous or not. We can also use one parameter to help us determine another—with math!

If we further break down the cell potential, it really looks like this:

$$ E = \frac {potential~difference~(J)} {charge~(C)} $$

We can further simplify this by classifying that potential difference = w (work) and charge = q. This gives us:

$$ E = \frac {w} {q} $$

By rearranging the formula and considering the work function, we arrive at:

$$ w = -qE^{\circ}_{cell} $$

If we remember that Faraday's Law is related to charge and number of electrons (n), we can substitute it into our q function. Recall Faraday's Constant, which represents the charge of electrons (in C) per mole:

$$ F=96485~C~mol^{-1} $$

\begin{align} q &= nF \\ w &= -qE^{\circ}_{cell} \\ w &= -nFE^{\circ}_{cell} \end{align}

Now that we've come to our work function, we can relate this to our free energy. When dealing with the maximum amount of work on a system, we can exchange the term for Gibbs Free Energy. Since ΔG = w_max, we get:

$$ \Delta G^{\circ} = -nFE^{\circ}_{cell} $$

With this new formula, it is now fairly simple to determine the cell potential when we know the Gibbs free energy change and vice versa.

Cell Potential and Free Energy Changes

Now that we have a formula to change between cell potential and free energy, we can easily determine one from the other. If we know the values of one, it is a lot easier to calculate the others than it is to measure them.

Now that we know if a reaction will be spontaneous, we also would like to know whether it is at Equilibrium or not. We know that the process of Calculating Equilibrium Constant relates it to our Gibbs free energy.

$$ \Delta G = -RT \ln{K} $$

With our previous derivation, we can now relate the equilibrium constant to our cell potential.

\begin{align} -nFE^{\circ}_{cell} &= -RT\ln{K} \\ E^{\circ}_{cell} &= \frac {RT} {nF} \ln{K} \end{align}

With this formula, we can not only measure the spontaneity of a reaction, but we can also measure its progress.

If we assume that we are under standard conditions we can quickly simplify this.

Let's assign the values for our standard conditions:

• T (temperature) = 25 °C = 273.15 K (kelvin)
• R (ideal gas constant) = 8.314 J mol^-1 K^-1
• F = 96485 C mol^-1

By plugging these values in, we end up with:

$$ E^{\circ}_{cell} = \frac {0.0257~V} {n} \ln{K} $$

Determining changes in cell potential just became easier!

Comparing Cell Potential, Free Energy, and Equilibrium

Since we have a new parameter, equilibrium, we should understand what this is telling us. Well, we know that reactions are typically under Dynamic Equilibrium, where they will try to reach a steady state. Atoms and molecules want to be at equilibrium because being higher in energy is hard work!

If we think about spontaneous reactions, we now know that it means that the forward reaction is probable, while the reverse reaction is improbable. How does that tie into equilibrium? Well, in a Dynamic Equilibrium, the equilibrium is always shifting back and forth slightly. In a spontaneous reaction, this means that it is more favored to go in the forward direction. So while it is always shifting, it is shifting more forward than it is backward.

Perhaps this table will help you visualize this.

The Relationship Between Standard Cell Potential and Free Energy

Now that we have derived the equation relating Gibbs free energy and cell potential, it should be noted that we have been dealing with the "standard" cell potential. Standard conditions are always denoted with a degree symbol (°). So, when you have seen E°_cell, or just E°, that means that the cell is under standard conditions.

If there is no degree symbol indicated, such as E, then you can assume the cell is not under standard conditions. This could be anything, such as a change in cell conditions, or if we are dealing with an electrolytic cell, which requires a voltage to operate.

Cell Potential and Free Energy Example

We have seen what cell potential and free energy are and how to calculate them. Let's take a look at an example to bring this all to life.