Free Energy and Electrochemical Potential

Electrical, osmotic, and chemical energies can perform work by directing the movement of a body against opposing forces. The quantitative measure of this energy conversion is the change in free energy. However, thermal energy at a constant temperature cannot perform work. In liquid-phase chemical reactions, pressure remains constant while volume may change. Therefore, for such systems, we consider the change in enthalpy (ΔH), defined as ΔU + pΔV (where p is pressure and ΔV is the change in volume), instead of the internal energy change. According to the first and second laws of thermodynamics, the relationship between the change in free energy (ΔG) and the change in enthalpy (ΔH) at constant pressure and temperature is given by:

ΔG = ΔH - TΔS

where ΔG is in Joules (J), ΔH is in Joules (J), T is in Kelvin (K), and ΔS is in Joules per Kelvin (J/K).

A negative ΔG indicates a spontaneous process, meaning the reaction will proceed without additional energy input. Conversely, a positive ΔG indicates a nonspontaneous process, requiring energy input to proceed.

In physicochemical systems, the change in free energy is typically described by the change in electrochemical potential (μ):

ΔG = m Δμ

where ΔG is in Joules (J), m is the amount of substance in moles (mol), and Δμ is in Joules per mole (J/mol).

The change in electrochemical potential when transitioning from state 1 to state 2 is determined by chemical, osmotic, and electrical energy changes:

Δμ = μ₂ - μ₁ + RT ln (c₂/c₁) + zF (φ₂ - φ₁)

where Δμ is in Joules per mole (J/mol), μ₁ and μ₂ are the initial and final chemical potentials in Joules per mole (J/mol), R is the gas constant (8.314 J/(mol·K)), T is temperature in Kelvin (K), c₁ and c₂ are the concentrations in moles per liter (mol/L), z is the charge number of the ion, F is the Faraday constant (9.65 × 10⁴ C/mol), and φ₁ and φ₂ are the initial and final electrical potentials in Volts (V).

The change in electrochemical potential signifies the work required to:

1. Synthesize 1 mole of a substance (state 2) from initial substances (state 1) and place it in the solvent (μ₂ - μ₁).
2. Concentrate the solution from concentration c₁ to c₂ (RT ln (c₂/c₁)).
3. Overcome electrical repulsion due to a potential difference (φ₂ - φ₁) between solutions (zF (φ₂ - φ₁)).

These terms can be either positive or negative.

Consider the transfer of sodium ions (Na⁺) through a nerve cell membrane as an example. This process is facilitated by the enzyme Na⁺, K⁺-ATPase and driven by ATP hydrolysis. Sodium ions move from the cell's interior to its exterior. The concentration of Na⁺ inside the cell (c₁) is 0.015 mol/L, while outside (c₂) it is 0.15 mol/L. The osmotic work for each mole of transferred ion at 37°C (310 K) is:

RT ln (0.15/0.015) = 8.314 J/(mol·K) × 310 K × ln (0.15/0.015) = 5.9 kJ/mol

Inside the cell, the electrical potential (φ₁) is -60 mV (-0.060 V), with the external potential (φ₂) set to 0 V. The electrical work is:

zF Δφ = 1 mol × 9.65 × 10⁴ C/mol × 0.060 V = 5.8 kJ/mol

Since no chemical transformations occur during the transfer and the ion remains in the same aqueous environment, Δμ₀ = 0. Therefore:

Δμ = 0 + 5.9 kJ/mol + 5.8 kJ/mol = 11.7 kJ/mol

Since Δμ is positive, the process of transferring sodium ions (Na⁺) through the nerve cell membrane is nonspontaneous. This means that it requires an input of energy, which in this case is provided by the hydrolysis of ATP, to proceed.