Enthalpy, Helmholtz and Gibbs Free Energy Explained

Enthalpy, Helmholtz free energy and Gibbs free energy are three special energy accounts used in thermodynamics to predict whether a process will happen on its own. Enthalpy tracks heat at constant pressure, while the two free energies tell us how much useful work we can get from a system.

These ideas matter because nature does not follow total energy alone. A reaction can release heat yet not occur, or absorb heat yet occur spontaneously. Free energies combine energy and entropy to give the correct answer at constant temperature and pressure.

Key Concepts

Internal energy (U) is the total kinetic and potential energy of particles inside a system, measured in joules (J). Enthalpy (H) is defined as H = U + PV, where P is pressure in pascals (Pa) and V is volume in cubic metres (m3). At constant pressure, the change in enthalpy equals heat absorbed: ΔH = Qp.

Helmholtz free energy (F) is F = U − TS, for constant temperature and volume. Gibbs free energy (G) is G = H − TS, for constant temperature and pressure. Spontaneity requires ΔF < 0 or ΔG < 0 respectively. T is in kelvin, S in J/K.

Formulas

H = U + PV. Change: ΔH = ΔU + PΔV at constant pressure. ΔH negative means exothermic (heat released), positive means endothermic.

F = U − TS. Change at constant T: ΔF = ΔU − TΔS. The maximum useful work obtainable at constant T and V equals −ΔF.

G = H − TS. Change at constant T and P: ΔG = ΔH − TΔS. This is the most used form. If ΔG < 0 the process is spontaneous, if ΔG > 0 it needs external work, and if ΔG = 0 the system is at equilibrium. Units are joules or kilojoules per mole (kJ/mol) in chemistry.

Worked Example

A reaction at 298 K has ΔH = −100 kJ/mol and ΔS = −0.200 kJ/(mol·K) (entropy decreases). Find ΔG and state if it is spontaneous.

Step 1: Use ΔG = ΔH − TΔS. Step 2: TΔS = 298 × (−0.200) = −59.6 kJ/mol. Step 3: ΔG = −100 − (−59.6) = −100 + 59.6 = −40.4 kJ/mol. Step 4: ΔG is negative, so the reaction is spontaneous at 298 K despite entropy decrease, because heat release dominates.

Common Misconceptions

• Misconception: Negative ΔH alone guarantees spontaneity. Correction: Spontaneity depends on ΔG. An endothermic process with large positive ΔS, like ice melting above 0°C, can be spontaneous.
• Misconception: Helmholtz and Gibbs energies are interchangeable. Correction: Use F for constant volume (sealed rigid container) and G for constant pressure (open beaker). Most lab and biological processes are at constant pressure, so G is used more.
• Misconception: Free energy is energy that is free or costless. Correction: It is the portion available to do useful work after accounting for entropy changes; the rest is tied up as unavailable heat.

Real-World Applications

Chemists use ΔG to predict whether a synthesis will work, find equilibrium constants via ΔG° = −RT ln K, and design batteries where ΔG gives the cell voltage: ΔG = −nFE. Biologists use Gibbs energy of ATP hydrolysis (about −30.5 kJ/mol) to explain how cells power muscles. Engineers use enthalpy in steam tables and combustion, and metallurgists use free-energy diagrams to extract metals from ores.

In short: enthalpy measures heat at constant pressure, Helmholtz energy gives useful work at constant volume, and Gibbs energy predicts spontaneity at constant temperature and pressure — together they turn energy bookkeeping into reliable predictions.