Well, now imagine you had rocket shoes. You'd be able to push that boulder up the hill no problem. That is precisely what Electrolysis deals with. Take an impossible reaction that would never happen on its own, apply a little voltage, and voilà, it works! Somehow this isn't a Myth, it's just chemistry.

But then the question becomes: how do we know if it will work on its own? And if not, how can we make it work? Well, all of those questions can be answered by understanding a little concept known as **cell potential**. What is this mythical value, and how do we figure it out? Well, stick around and find out!

- First, we will discuss
**cell potential**: what it means, what it looks like, and how we calculate it. - Then, we'll take a look at standard operating conditions, and what happens when conditions are no longer theoretical.
- Finally, we'll take a look at an example to lock all this learning in.

## Cell Potential Meaning

If you place a ball at the top of a steep hill, what will happen? Well, if the ball is placed on a flat spot, it might just stay there, waiting for some force to push it so that it can roll down the hill. By rolling down the hill, it will generate kinetic energy (the energy of motion) and accelerate down the hill. But before it starts to roll, it will possess potential energy, a kind of stored energy. At the very top, it has a high amount of potential and at the bottom, it has a low amount of potential to roll.

This concept of potential energy is observed in every aspect of nature, but more specifically, in electrochemical cells. If we have a setup of Galvanic and Electrolytic Cells, each cell will contain two different species: an **oxidizing** species and a **reducing** species.

An **oxidizing** species—or **oxidant**—will oxidize another species by taking electrons away from it. A **reducing** species— or **reductant**—will reduce another species by giving it electrons.

By setting these two cells up together, one cell will try to give electrons to the other. You can imagine this as a flow of electrons, or as kinetic energy. Considering our previous analogy, imagine that one chemical cell is at the top of the hill and one is at the bottom. The higher cell is trying to push electrons (the ball) toward the cell at the bottom, where there is a lower potential energy.

The difference in energy between these two half-cells is called a **potential difference**. When we consider Galvanic (a.k.a. Voltaic) cells, we consider a property called **cell potential**.

**Cell potential** is the potential difference between two electrodes measured in volts (V).^{1}

So, a high cell potential, means that one electrode *really* wants to transfer electrons to the other. If allowed, this reaction will occur, and you will observe a flow of electrons. But what about the other direction? Can you roll the ball back up the hill? All is possible through the use of Electrolysis, but it is important to know whether a reaction will occur spontaneously or not.

### Measuring Cell Potential

It is important to briefly discuss how measuring cell potential is performed. Well, by now you know that it is measured in Volts, but how is this done? Why, with a voltmeter, of course! A typical electrochemical cell is shown here.

As is typical in electrochemical cells, the anode will send electrons to the cathode, which will generate current. Under standard conditions, the value of a cell's potential is very predictable. Many scientists over the past century have determined the standard cell potential of numerous redox reactions. This means, if we hook up any two half cells together, we know whether they will react or not.

Now wait, what do we mean by, "if they will react or not"? Remember that a cell will only generate current if there is a potential difference, which means the anode has to be higher in potential than the cathode. So, essentially, if we hook them up the wrong way, there won't be any flow of electrons.

This is always the case with Galvanic and Electrolytic Cells. A galvanic (Voltaic) cell describes one that has electrons go from a high potential difference to a low one. Or, in other words, a cell that hosts a spontaneous reaction.

The opposite direction is also possible through the power of electricity. An electrolytic cell is one that works at negative cell potentials by applying a voltage.

## Cell Potential Equation

Each reduction reaction (half cell) has its own potential value. Meaning, you can pick any two cells you want and determine how much cell potential, *E°*_{cell}, is needed to power the reaction. This is done through the cell potential equation.

A positive value means that the reaction is spontaneous and will occur on its own. A negative potential means that we need to crank the voltage to make up for the rest. At this point, we've talked a lot about using half-cell potentials to calculate whether a reaction will work, so let's get some practice.

Say we've selected zinc and copper as our two electrodes which we want to react. We would like to figure out whether, when combined, they will have a positive cell potential. What we do first is consult a reduction half-reaction table. This is a table that you can find in any chemistry textbook that discusses electrochemistry. Here is a simplified version of one:

Oxidized form + electrons | $\underset{}{\to}$ | Reduced form | E° (V) |

$$ O_2 (g) + 4H^+ + 4e ^- $$ | $\underset{}{\to}$ | $$ 2 H_2O(l) $$ | 1.23 |

$$ Cu^{2+} (aq) + 2 e^- $$ | $\underset{}{\to}$ | $$ Cu (s) $$ | 0.34 |

$$ 2 H^+ (aq) + 2 e^- $$ | $\underset{}{\to}$ | $$ 2 H_2O (l) $$ | 0 |

$$ Zn^{2+} (aq) + 2 e^- $$ | $\underset{}{\to}$ | $$ Zn (s) $$ | -0.76 |

$$ Al^{3+} (aq) + 3 e^- $$ | $\underset{}{\to}$ | $$ Al (s) $$ | -1.66 |

Even in this small, simplified version of a reduction table, there is a lot of information. The first important thing to note is that every reaction is written as a reduction. Notice that in every single reaction, the reactants are gaining electrons and are being reduced. The second thing to note is that each reduction has its own unique *E°* value.

And the third thing is that the values go in order from positive to negative. This is referred to as each cell's reduction potential. The higher the half-cell potential, the stronger the oxidizing agent. Conversely, the lower the value (more negative), the stronger the reducing agent. In other words, the more positive half-cell potential will try to *take* electrons, and the more negative value will try to *give* electrons.

Okay, back to our example of zinc and copper. Now that we have our half-cell potentials, we can introduce an equation to determine the overall cell potential.

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

Okay, so how do we know which half-cell will be the cathode and which will be the anode? Well, now it is time to combine the stuff we've just learned.

So, we *know* that the anode will be trying to send electrons to the cathode, since electrons will flow downhill (remember the ball analogy?). We also *know* that a more positive half-cell potential will be taking electrons from the more negative cell. Let's make zinc our anode, and copper our cathode.

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

So, what does this mean? Well, our overall cell potential is a positive value, which means that this is a spontaneous reaction! But, just to make sure, let's try putting it the other way around just to see what would happen.

\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} = -0.76~V - (0.34~V) \\&E^{\circ}_{cell} = -1.10~V\end{align}

Since it is a negative value, we know it won't work, just like we know how water won't flow uphill, and that boulder is never going to reach the top of that mountain. Not unless we use an external power source and rocket it up there.

### Converting Cell Potential to Gibbs Free Energy and Equilibrium

The cell potential can also be used to help determine the Gibbs Free Energy change of a system and even the position of Equilibrium in a system. For a detailed derivation of converting cell potential to Gibbs free energy and equilibrium, be sure to check out Cell Potential and Free Energy. Here we will just show the relationship between them.

Each of the three variables are related through thermodynamics, which is why they can be converted. These values and equations are used for standard conditions. When we encounter systems that are no longer under standard conditions, it changes some assumptions, which means we need different formulas.

## Cell Potential Conditions

The cell potential conditions depend on what type of electrochemical cell you are dealing with. In all cases, the anode will be giving electrons to the cathode. However, a Galvanic cell will have a positive cell potential and an electrolytic cell will have a negative cell potential.

Next, it comes down to standard versus nonstandard conditions. But, what does that even mean? Well, in the lab, we generally try to keep conditions constant to make sure that our results are 1) as accurate as possible, and 2) reproducible. These conditions refer to simple things like temperature, pressure, concentration,~~ and ~~~~good music~~.

Unfortunately, if we aren't using perfect equipment, there will inevitably be some measure of error. To help account for this, it becomes relevant to introduce an equation that is meant for nonstandard cell potential conditions.

### Nonstandard Conditions

We saw earlier that cell potential under standard conditions can be related to the equilibrium constant. Remember that when a state is not at equilibrium, the Reaction Quotient is used instead. This term can be applied to find the change in Gibbs free energy *(ΔG)* under nonstandard conditions. The following relationship can be derived for the system:

$$ \Delta G = \Delta G^{\circ} + RT \ln {Q} $$

- R = ideal gas constant (J mol
^{-1}K^{-1}) - T = temperature (K)
- Q = reaction quotient

When writing units, there are a few ways to do so. It is important to be able to recognize each method. The universal gas constant has units of \( \frac {J} {(mol \times K)}\), or \(J \cdotp mol^{-1} \cdotp K^{-1}\), or \(J~mol^{-1}~K^{-1}\). All are acceptable, so make sure you are familiar with each method.

We saw earlier in Cell Potential and Free Energy that standard cell potential can be derived from standard Gibbs free energy. Well, we will use a similar relationship here:

\begin{align}\Delta G &= \Delta G^{\circ} + RT \ln {Q}\\ -nFE_{cell} &= -nFE^{\circ}_{cell} + RT \ln {Q} \\E_{cell} &= E^{\circ}_{cell} - \frac {RT} {nF} \ln {Q}\end{align}

After some derivation (which will not be discussed here), we arrive at a simplified version of this formula.

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

This equation is absolutely fundamental in electrochemistry and is known as the **Nernst equation**. When trying to determine the cell potential under nonstandard conditions, the Nernst equation is the go-to formula. It is especially useful for determining cell potential in situations where the half-cells contain different concentrations of electrolyte.

Nonstandard conditions are important to understand since they are often prevalent outside of lab conditions.

$$ H_2O (l) \rightleftharpoons H_2O (g) \qquad \Delta G^ {\circ} = 8.59 ~ kJ ~ mol ^{-1} $$

This reaction states that under standard conditions, the evaporation of water is *not* thermodynamically favored.

So, if you were to spill some water on the ground, it would never evaporate. In fact, more water would condense along with it because the reverse reaction *is* thermodynamically favored. We know that this is not the case, and that would likely never happen unless the air was supersaturated with water.

We can generally assume unless otherwise stated, that the conditions we are dealing with are not standard. However, a deeper dive into nonstandard potential will be saved for Non-standard Conditions.

## Cell Potential Chart and Table

The Nernst equation describes cell potential and its relationship with equilibrium. Essentially, there are consequences for whether there is a higher concentration of reactants, products, or if they're the same. This may be difficult to conceptualize, so it is generally useful to craft a cell potential chart and table

\( \bf Q = x \) | \( \bf E _ {cell} = y \) |

\( Q \gt 1 \) | \( E _ {cell} \lt E^ {\circ} _ {cell} \) |

\( Q \lt 1 \) | \( E _ {cell} \gt E^ {\circ} _ {cell} \) |

\( Q = 1 \) | \( E _ {cell} = E^ {\circ} _ {cell} \) |

\( Q = K \) | \( E _ {cell} = 0 \) |

So, you may look at this table and understand everything. Or, you may be looking at it, and it just seems like random values that mean nothing to you. If you're the second person, that is totally okay. Let's break it down together to put it into words.

What this table says is that the nonstandard cell potential will be determined based on the position of equilibrium. If the concentration of products is *higher* than the reactants, the cell potential will be *lower* than the standard potential.

So, essentially, if the cell potential is lower than the standard, the reaction will want to go in reverse. This is what we saw in the previous example. The opposite can be said too. If the potential is higher than the standard, then the reaction will go in the forward direction.

Well, what about if the products and reactants are equal? This means that the cell is under standard conditions and the cell potential is just the standard potential. And finally, if the reaction reaches equilibrium, there will be no cell potential because there will be no difference in potential energy. Does that all make sense? If not, try plugging these values into the Nernst equation and see what happens.

## Cell Potential Symbols

Sometimes it can be challenging to remember all the different variables, units, and cell potential symbols. So, here is a table for you in case you forget what each one is.

Symbol | Name | Units |

E_{cell} | Overall cell potential | V (volts), J C^{-1} (Joules / Coulomb) |

E°_{cell} | Standard cell potential | V (volts), J C^{-1} (Joules / Coulomb) |

ΔG | Change in Gibbs free energy | kJ mol^{-1} (kilo Joules) |

ΔG° | Change in standard Gibbs free energy | kJ mol^{-1} (kilo Joules / mole) |

Q | Reaction quotient | Unitless |

K | Equilibrium constant | Unitless |

n | Number of electrons exchanged | Unitless |

F | Faraday's constant | C mol^{-1} (Coulomb / mole) |

R | Ideal gas constant | J K^{-1} mol^{-1} (Joule / Kelvin * mole) |

T | Temperature | K (Kelvin), °C (degrees Celsius) |

Note that this is not an all-encompassing list, and you may encounter others. Furthermore, there are other units that these values can have.

Hopefully, by now you have a better understanding of cell potential. Remember that with a reduction potential table, and the Nernst equation, you can determine the spontaneity of any electrochemical cell you can think of. Try some out and fulfill your destiny as the next, best, electrochemist!

## Cell Potential and Nonstandard Conditions - Key takeaways

**Cell potential**is the difference in potential energy between two half cells, expressed with the formula:\( E^{\circ}_{cell} = E^{\circ}_{cathode} - E^{\circ}_{anode} \)- Under
**nonstandard conditions,**we can use the formula:\( E_{cell} = E^{\circ}_{cell} - \frac {RT} {nF} \ln {Q} \) - If the temperature is set to 25 °C, we use the
**Nernst****equation**:\( E_{cell} = E^{\circ}_{cell} - \frac {0.0257~V} {n} \ln {Q} \) - An overall cell potential that is greater than the standard cell potential will be more thermodynamically favored. An overall cell potential which is lower than the standard cell potential will be less thermodynamically favored.

## References

- Nivaldo Tro, Travis Fridgen, Lawton Shaw,
*Chemistry a Molecular Approach*, 3rd ed., 2017 - Fig. 1 - Galvanic cell ( https://glossary.periodni.com/glossary.php?en=galvanic+cell) by E. Generalic is licensed by CC BY 4.0.

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##### Frequently Asked Questions about Cell Potential

What is cell potential?

**Cell potential** is the potential energy difference between two half cells in a Galvanic or electrolytic cell. It tells if a redox reaction will be thermodynamically allowed.

What is the cell potential formula?

The formula for cell potential under nonstandard conditions is the Nernst equation:

\( E_{cell} = E^{\circ}_{cell} - \frac {0.0257~V} {n} \ln {Q} \)

What is standard potential of a cell?

The standard potential is the thermodynamic favourability of a reaction under standard conditions. It is determined with the formula:

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

What is cell potential and electrode potential?

Electrode potential is the cell potential of a half cell which is compared against a standard electrode. This is done to determine the reduction potential of a specific reaction.

What are the measures of cell potential?

Cell potential is the product of the potential difference (J) divided by the charge (C). It has the units of J C^{-1}, often just indicated as volts (V).

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