## ME 527: Lesson 14 – All About Gibbs!

Elijah Shoemake## Discovery and Origins – J. Willard Gibbs

Josiah Willard Gibbs, born in 1839, was deemed by Einstein as “the greatest mind in American history.” Gibbs primarily focused on physical chemistry and its chemical reaction relationship with thermodynamics. During the years 1876-1878, he published the famous “On the Equilibrium of Heterogeneous Substances” in the Transactions of the Connecticut Academy of Arts and Sciences. He will later earn the most prestigious international science award in its time, the “Copley Medal of the British Royal Society” in 1901. Within the publication, the discovery of free energy emerged; now known as Gibbs free energy1. Gibbs described this energy as:

“The greatest amount mechanical work which can be obtained from a given quantity of a certain substance in a given initial state, without increasing its total volume or allowing heat to pass to or from external bodies, except such as at the close of the processes are left in their initial condition.”

This “mechanical work” or energy can be thought of as the maximum amount of non-expansion work extracted from a closed system. The work done is at the expense of the internal energy of the system and any portion of the work not extracted is lost to the environment in the form of heat.

His discovery led to the development of quantum mechanics, further discoveries in theoretical physics, and in statistical mechanics.

## Gibbs Free Energy

Gibbs free energy is usually expressed as:

Where h is the enthalpy, T is the temperature, and s is the entropy, P is the pressure, and v is the specific volume.

The Gibbs fundamental property relation equation (dg) shows the most commonly used form of the Gibbs Free Energy Equation. Gibbs is most useful being in terms of temperature and pressure. This is because the ability to quantify entropy and -enthalpy pose to be a much more complicated task.

Although the Helmholtz Free Energy Equation is the most used fundamental property relation, Gibbs’ equation is still used primarily by chemists and physicists. Its primarily uses are to determine:

- The chemical potential
- Electrochemical properties of fluids
- Excess enthalpies of fluid mixtures
- Equilibrium properties of fluids when the temperature and pressure of the system is known

The dg predicts the direction of the chemical reaction and tells whether the reaction is spontaneous or nonspontaneous; meaning, whether the reaction may occur with or without applied external energy to the system. Note that a negative dg between two phases does not guarantee that a phase change will happen quickly, or even at all! At atmospheric temperature and pressure a diamond is NOT the lowest Gibbs energy state, however metastability means that it will take a time that is comparable with the duration of the universe before it will turn back into its Carbon form. The same can be seen in supercooled liquids below the freezing point, where an impurity or energy inputs can cause it to rapidly freeze. The use of a catalyst, impurities, or seed crystals can significantly lower the dg necessary to instigate a phase change. This is an important concept in interfacial phenomena where one can decrease, or even increase the energy barrier necessary to casue a phase change.

Although Gibbs’ equation is classified as an Equation of State, Gibbs’ equation is not the ideal fundamental property relation to use in plotting specific states; such as, density, pressure, temperature, etc.; s discussed in Lesson 10. This is because the Gibbs equation cannot accurately determine specific states when the temperature and pressure are not independent quantities; predominately at phase changes. As Dr. Leachman stated, the “inability of the Gibbs Energy to differentiate phase in the most commonly used temperature and pressure coordinates is why modern EOS are explicit in Helmholtz Energy.2”

Below is the correlation between the commonly used form of Gibbs Free Energy and the most used form of the Helmholtz Energy Equation:

The Helmholtz Equation can be represented by:

The most common form of the Helmholtz Equation is in its reduced non-dimensionalized form:

The relation of Gibbs to Helmholtz energy equation can then be represented as:

Where α is the reduced Helmholtz energy and:

where tao is the reduced inverse temperature and delta is the reduced density. Typically, the critical density and temperature are used in place of the the reduced density and temperature. This is done to reduce the density and the inverse temperature.