Cell potential (\(E_{cell}\)) represents the voltage, or driving force, of an electrochemical reaction involving electron transfer. Determining the overall voltage an electrochemical cell can produce often leads to confusion regarding the mathematical formula. Specifically, the question arises whether the calculation involves adding or subtracting the potentials of the two half-reactions. This article clarifies the standard method for calculating cell potential and explains what the resulting value signifies about the cell’s operation.
Understanding the Roles of Anode and Cathode
An electrochemical cell consists of two half-cells, each with an electrode where chemical reactions occur. These electrodes are designated as the anode and the cathode, differentiated by the chemical process at their surfaces.
The anode is the electrode where oxidation occurs, involving the loss of electrons. These released electrons flow out of the anode and into the external circuit. Conversely, the cathode is the electrode where reduction takes place, involving the gain of electrons. Electrons flowing through the external circuit arrive at the cathode to be consumed in the reduction half-reaction.
This flow of electrons from the anode to the cathode generates the electrical current. Regardless of the cell’s charge designation (positive or negative), the anode is always the site of oxidation, and the cathode is always the site of reduction. The electrolyte solution connects the half-cells internally, allowing ions to move and maintain charge neutrality as electrons move through the external circuit.
The Standard Rule for Calculating Cell Potential
The overall cell potential (\(E_{cell}\)) is a measure of the difference in electrical potential between the cathode and the anode. To calculate this voltage, the standard convention dictates the use of standard reduction potentials (\(E^\circ\)). These are pre-measured values for half-reactions under standard conditions, such as 1.00 M concentration and 25°C temperature. These values are all measured relative to the Standard Hydrogen Electrode (SHE), which is assigned an \(E^\circ\) of 0.00 V.
The widely accepted formula for calculating the standard cell potential is:
\(E^\circ_{cell} = E^\circ_{reduction, cathode} – E^\circ_{reduction, anode}\)
This method utilizes the tabulated standard reduction potentials directly for both the cathode and the anode half-reactions. This is done without manually flipping the sign for the oxidation reaction at the anode. The subtraction inherently accounts for the fact that the reaction at the anode is an oxidation, even though its potential is listed as a reduction potential.
The reason this subtraction works is that the standard reduction potential of the species that undergoes oxidation (the anode) is inherently lower, or more negative, than the potential of the species that undergoes reduction (the cathode). Subtracting a smaller, or more negative, number from a larger, or more positive, number results in a positive \(E^\circ_{cell}\) for a spontaneous reaction.
For example, if a copper half-cell (\(E^\circ = +0.34\) V) is the cathode and a zinc half-cell (\(E^\circ = -0.76\) V) is the anode, the calculation is \(E^\circ_{cell} = (+0.34\) V) – (-0.76$ V) = +1.10$ V. By consistently using the \(E^\circ_{cathode} – E^\circ_{anode}\) formula, the calculation remains straightforward. The potential difference calculated is the maximum electrical work the cell can perform, measured in volts.
Interpreting the Result: Cell Spontaneity and Type
The sign of the calculated cell potential (\(E_{cell}\)) determines the spontaneity and fundamental nature of the electrochemical reaction.
Positive Cell Potential (Galvanic Cells)
A positive value for \(E_{cell}\) indicates a spontaneous reaction under the specified conditions. A spontaneous reaction proceeds without continuous external energy input, meaning the cell generates electrical energy from a chemical reaction. Cells with a positive \(E_{cell}\) are classified as Galvanic or Voltaic cells, which are the type commonly used as batteries. In these cells, the chemical energy stored in the reactants is converted into electrical energy, which can then be used to do work in an external circuit.
Negative Cell Potential (Electrolytic Cells)
Conversely, a negative value for \(E_{cell}\) signifies a non-spontaneous reaction. This means the chemical process will not occur on its own and requires an external source of electrical energy to proceed. Cells operating under these conditions are called Electrolytic cells.
The connection between cell potential and spontaneity is formally established by the relationship with Gibbs Free Energy (\(\Delta G\)), which is a thermodynamic quantity that also predicts spontaneity. The equation linking the two is \(\Delta G = -nFE_{cell}\), where \(n\) is the number of moles of electrons transferred, and \(F\) is Faraday’s constant. Since \(n\) and \(F\) are always positive values, a positive \(E_{cell}\) must correspond to a negative \(\Delta G\). A negative \(\Delta G\) is the thermodynamic condition for a spontaneous process, reinforcing the conclusion that a positive \(E_{cell}\) indicates a power-generating cell. This simple sign convention, derived from the standard calculation method, allows scientists and engineers to quickly determine whether a particular cell design will act as a power source or require a power source for its operation.