Gibbs Energy and Equilibrium Constant

[BioPhysicDino]

The relation, ΔG = -RT ln(K_eq)
is ubiquitous.

It says that for a reaction, the change in Gibbs energy is proportional to the logarithm of the equilibrium constant.

But where does this come from?
I've been reading through many books and haven't yet find any that derive or indicate the history of this equation. Does anyone know?

Is this simply an empirical result? Is it derivable other than from statistical mechanics? I've found a couple places where this can be derived from statistical mechanical postulates, but is that the only way? Is that where this relation originally comes from? (I thought it was originally based in thermodynamics; am I wrong about that?)

Does anyone know the history of this equation? Who first wrote it down? In what publication?

If you can shed some light in where this relationship originally came from I would much appreciate it. Thank you. --Dino

 

[PhaseShifter]

I'm pretty sure it was rigorously derived for ideal gases. Pardon my rambling while I try to remember what I can...
dU=dq+dw=TdS-pdV
G=U-TS+PV,
so dG= VdP-SdT
Since V=nRT/P, this gives dG=nRT d(ln P) - S dt
Dividing by n to get molar quantities,

$$dG=RT d(ln P)-S dT$$

For a constant-temperature process we can drop that last term.
so for a single component, we can integrate both sides to get the change in ΔG when we expand or compress a gas isothermally:

$$G_{actual}-G_{standard state}=RT ln{{P}\over{P_{standard}}}$$

Now combine that for all products and reactants in a reaction, and set standard pressure equal to 1 atm and you'll get

$$\Delta G_{actual}-\Delta G_{standard state}=RT ln{q}$$

Since the actual ΔG at equilibrium is zero (otherwise it wouldn't be at equilibrium), and $$q=K_{eq}$$ at equilibrium this simplifies to

$$-\Delta G_{standard state}=RT ln{K_{eq}}$$

or the more famous

$$\Delta G_{standard state}=-RT ln{K_{eq}$$

Ideal solutions should behave similarly (replacing pressure with concentration, although I couldn't derive it at the moment).

 

[quicknote]

The relationship between Gibbs energy and equilibrium constant is a fundamental concept in thermodynamics and was first described by American physicist Josiah Willard Gibbs in the late 19th century. It is derived from the second law of thermodynamics, which states that in a closed system, the total entropy (a measure of disorder) will always increase over time. This law also applies to chemical reactions, where the system is the reactants and products, and the total entropy can be calculated using the change in Gibbs energy.

The equation ΔG = -RT ln(Keq) is derived from the Gibbs free energy equation, which relates the change in Gibbs energy to the change in enthalpy (heat) and entropy of a system. This equation is based on statistical mechanics, which uses statistical methods to describe the behavior of a large number of particles. It can also be derived from thermodynamic principles, making it a robust and widely applicable relationship.

The equilibrium constant, Keq, is a measure of the ratio of products to reactants at equilibrium and is related to the Gibbs energy change by the equation ΔG = -RT ln(Keq). This relationship is ubiquitous because it allows us to predict the direction and extent of a chemical reaction at equilibrium. It also shows the dependence of equilibrium on temperature, as the value of Keq changes with temperature due to the influence of entropy.

The history of this equation can be traced back to Gibbs' work on thermodynamics and statistical mechanics in the late 1800s. It has since been refined and expanded upon by numerous scientists, including Austrian chemist Ludwig Boltzmann and German physicist Max Planck. Today, it is a fundamental concept in chemistry and is used in various fields, from industrial processes to biological systems.

In summary, the relationship between Gibbs energy and equilibrium constant is a well-established and fundamental concept in thermodynamics, derived from both statistical mechanics and thermodynamic principles. Its history can be traced back to the work of Josiah Willard Gibbs and has since been refined and expanded upon by other scientists.