The voltage of a battery is fundamentally determined by the electrochemical potentials of its constituent electrodes. This relationship is rooted in thermodynamics and electrochemistry, where standard electrode potentials (E°) serve as the basis for calculating the theoretical voltage of an electrochemical cell. The potential difference between the anode and cathode, when both are under standard conditions, defines the cell's electromotive force (EMF). This EMF is a direct consequence of the Gibbs free energy change associated with the redox reactions occurring at each electrode.
Standard electrode potentials are measured relative to the standard hydrogen electrode (SHE), which is arbitrarily assigned a potential of 0.000 V under standard conditions (298 K, 1 atm pressure, and 1 M solute concentration). The SHE consists of a platinum electrode in contact with hydrogen gas at 1 atm and an acidic solution with a hydrogen ion activity of 1. When another half-cell is connected to the SHE, the resulting cell potential provides the standard reduction potential for that half-reaction. For example, if a zinc electrode immersed in a 1 M Zn²⁺ solution is paired with the SHE, the measured cell voltage is -0.763 V, indicating that zinc has a lower reduction potential than hydrogen.
The overall cell potential (E°cell) is calculated by subtracting the standard reduction potential of the anode from that of the cathode:
E°cell = E°cathode - E°anode
This equation assumes that both half-reactions are written as reductions. If one of the reactions is an oxidation, its sign must be reversed before performing the subtraction. The resulting E°cell represents the maximum voltage the cell can deliver under standard conditions when no current is flowing.
The Nernst equation extends this concept to non-standard conditions by accounting for variations in temperature, pressure, and concentration. It relates the cell potential (Ecell) to the standard cell potential (E°cell) and the reaction quotient (Q):
Ecell = E°cell - (RT/nF) ln Q
Where:
- R is the universal gas constant (8.314 J/mol·K)
- T is the temperature in Kelvin
- n is the number of electrons transferred in the redox reaction
- F is Faraday's constant (96,485 C/mol)
- Q is the reaction quotient, which reflects the instantaneous concentrations of reactants and products
At room temperature (298 K), the Nernst equation can be simplified using natural logarithm to base-10 conversion:
Ecell = E°cell - (0.0592 V/n) log Q
This form is particularly useful for calculating how cell voltage changes with varying ion concentrations. For instance, as a battery discharges, reactant concentrations decrease while product concentrations increase, causing Q to rise and Ecell to fall. The Nernst equation quantitatively describes this gradual voltage decline during discharge.
The connection between Gibbs free energy (ΔG) and cell potential is derived from thermodynamic principles. The maximum electrical work (Welec) a battery can perform is equal to the negative change in Gibbs free energy:
Welec = -ΔG
Since electrical work is also the product of charge transferred (nF) and cell potential (Ecell), the two expressions can be equated:
ΔG = -nFEcell
Under standard conditions, this becomes:
ΔG° = -nFE°cell
This relationship demonstrates that a positive E°cell corresponds to a negative ΔG°, indicating a spontaneous reaction. The more positive the cell potential, the greater the driving force for the redox reaction. Conversely, if E°cell is negative, ΔG° is positive, and the reaction is non-spontaneous under standard conditions.
The standard hydrogen electrode plays a pivotal role in determining these potentials. By serving as a universal reference, it allows for the systematic tabulation of reduction potentials for various half-reactions. These tabulated values enable the prediction of cell voltages for countless combinations of electrodes without direct measurement. For example, knowing that the standard reduction potential of Cu²⁺/Cu is +0.337 V and that of Zn²⁺/Zn is -0.763 V, one can immediately calculate that a Daniell cell (zinc anode, copper cathode) has a standard cell potential of:
E°cell = 0.337 V - (-0.763 V) = 1.100 V
The Nernst equation then allows adjustment of this value for non-standard concentrations. If the zinc ion concentration is 0.1 M while the copper ion concentration is 0.01 M, the cell potential at 298 K becomes:
Ecell = 1.100 V - (0.0592 V/2) log (0.1/0.01) = 1.100 V - 0.0296 V = 1.070 V
This decrease reflects the less favorable conditions compared to standard state.
The temperature dependence of cell potential is also captured by the Nernst equation through the RT/nF term. As temperature increases, the magnitude of this correction factor grows, meaning concentration effects become more pronounced at higher temperatures. This has practical implications for battery operation across different thermal environments.
The relationship between equilibrium constants and cell potential further demonstrates the thermodynamic foundation of battery voltage. At equilibrium, Ecell = 0 and Q = K (the equilibrium constant). Substituting into the Nernst equation yields:
0 = E°cell - (RT/nF) ln K
Rearranging provides a method to determine equilibrium constants from standard potentials:
E°cell = (RT/nF) ln K
This shows that reactions with large positive standard potentials have very large equilibrium constants, proceeding nearly to completion. In battery terms, this translates to highly energetic cells with strong tendencies to discharge.
Practical battery voltages often deviate from these ideal predictions due to internal resistance, overpotentials, and kinetic limitations. However, the fundamental thermodynamic relationships remain valid and provide the theoretical framework for understanding and designing electrochemical energy storage systems. The standard potentials establish baseline expectations, while the Nernst equation enables refinement of these predictions for real-world operating conditions.
The interplay between these concepts governs not just the voltage but also the energy capacity of batteries. Since the total energy is the product of voltage and charge (which depends on n), materials with higher reduction potentials and multiple electron transfers per formula unit offer greater energy densities. This explains why lithium-based systems dominate high-performance applications—their highly negative standard potentials (-3.040 V for Li⁺/Li) enable large cell voltages when paired with appropriate cathodes.
In summary, battery voltage originates from the difference in tendencies of materials to gain or lose electrons, quantified by standard electrode potentials. The Nernst equation bridges the gap between ideal standard conditions and practical operating states, while thermodynamic principles connect these electrical measurements to fundamental energy changes. Together, these relationships form the theoretical foundation for electrochemical energy storage, enabling the calculation and optimization of battery performance across diverse applications.