Work at Constant Volume
While Gibbs Free Energy is used for systems at constant pressure (like test tubes open to the atmosphere), Helmholtz Free Energy ($A$) is the thermodynamic potential used for systems at a constant volume and temperature.
Named after German physicist Hermann von Helmholtz, it measures the maximum amount of useful work that can be extracted from a closed system where the volume cannot expand or contract.
Constant Volume Physics
- Bomb Calorimeters: In chemistry, the explosive energy of food or fuel is measured inside a thick steel container called a bomb calorimeter. Because the steel walls cannot expand, the volume is constant, and the energy changes are strictly governed by Helmholtz free energy, not Gibbs.
- Explosives Engineering: When calculating the destructive potential of an explosive confined inside a closed casing or a rock bore-hole, engineers must use the Helmholtz potential. The work done to fracture the casing is entirely derived from the change in $A$.
- Statistical Mechanics: In advanced theoretical physics and computational chemistry, the Helmholtz free energy is mathematically much easier to derive directly from the partition function (the probability states of all molecules) than any other thermodynamic property.
The Formula
A = U - TS
Example Calculation
A sealed, rigid tank of pressurized gas has an internal energy ($U$) of $15,000 , \text{Joules}$ and a total entropy ($S$) of $40 , \text{J/K}$. The tank is kept in a temperature-controlled bath at $300 , \text{K}$.
- Multiply T and S: $300 \cdot 40 = 12,000 , \text{Joules}$.
- Subtract from U: $15,000 - 12,000 = 3,000 , \text{Joules}$.
The Helmholtz free energy is $3,000 , \text{Joules}$. This means if you let the pressurized gas slowly leak out and drive a turbine, the absolute maximum theoretical mechanical work you could ever extract is exactly $3,000 , \text{Joules}$, even though the internal energy is much higher.