Thermodynamics || Lec # 7 || Relationship b/w Change in Equilibrium Constant and Gibbs Free Energy

TL;DR

At equilibrium, the Gibbs free energy change satisfies ΔG = 0.

Briefing Cornell Notes

Briefing

Gibbs free energy links chemical equilibrium to measurable reaction constants: at equilibrium, the change in Gibbs free energy is zero, and that condition ties directly to the equilibrium constant through a pressure (or concentration) ratio. The lecture builds the relationship by starting from the thermodynamic expression for free-energy change in an ideal-gas reaction and then rewriting it in terms of equilibrium constants.

For a reaction that reaches equilibrium, the concentrations of reactants and products can be represented generically (for example, reactants A and B with concentrations a and b, and products C and D with concentrations c and d). The key step is expressing the change in free energy using a logarithmic dependence on the ratio of pressures (or concentrations) of products to reactants. For an ideal gas, the lecture uses the form involving natural logarithms of pressure terms, treating the pressure change from an initial reference (like 1 atmosphere) to an arbitrary pressure p. This allows the free-energy change to be written in terms of pressure ratios that can later be mapped onto equilibrium constants.

The lecture then clarifies the difference between free energy and standard free energy. Free energy is defined without fixing conditions, so it varies with temperature and pressure. Standard Gibbs free energy, by contrast, is measured at specified standard temperature and standard pressure, which makes it comparable across experiments and laboratories. When the reaction proceeds, the total change in Gibbs free energy can be expressed as the sum of contributions from products minus those from reactants, using the standard Gibbs free energies of each species.

Next comes the equilibrium condition. The lecture uses the sign of ΔG to classify reaction behavior: if ΔG is zero, the system is at equilibrium; if ΔG is negative, the reaction is spontaneous; if ΔG is positive, it is non-spontaneous. At equilibrium, setting ΔG = 0 collapses the thermodynamic expression into a relationship where the pressure (or concentration) ratio becomes equal to the equilibrium constant. In other words, the equilibrium constant emerges from the same logarithmic pressure-ratio term that appears in the Gibbs free energy expression.

Finally, the relationship is presented as a practical bridge: knowing the equilibrium constant at a given temperature and pressure lets one determine the Gibbs free energy for that reaction, since Gibbs free energy depends on temperature and pressure. The takeaway is that equilibrium constants are not just empirical numbers—they are thermodynamic fingerprints of how Gibbs free energy changes when reactants convert into products.

Cornell Notes

The lecture connects chemical equilibrium to Gibbs free energy through a logarithmic pressure (or concentration) ratio. Starting from an ideal-gas expression for free-energy change, it rewrites ΔG in terms of standard Gibbs free energies of reactants and products plus a term involving pressures of products divided by pressures of reactants. It distinguishes free energy (depends on the system’s actual temperature and pressure) from standard Gibbs free energy (defined at specified standard conditions). Using the equilibrium criterion—ΔG = 0—the pressure-ratio term becomes the equilibrium constant. This yields a direct thermodynamic link: equilibrium constants at a given temperature and pressure determine the Gibbs free energy change for the reaction.

How does the lecture use Gibbs free energy to decide whether a reaction is at equilibrium or not?

It uses the sign of ΔG: when ΔG = 0, the reaction is at equilibrium; when ΔG < 0, the reaction is spontaneous; and when ΔG > 0, the reaction is non-spontaneous. Since equilibrium corresponds to ΔG = 0, the equilibrium condition is applied to the derived ΔG expression to extract the equilibrium-constant relationship.

What is the practical difference between free energy and standard Gibbs free energy in this lecture?

Free energy is not tied to fixed conditions, so it changes with the system’s temperature and pressure. Standard Gibbs free energy is measured at specified standard temperature and standard pressure, making it consistent and comparable across experiments. That fixed-condition definition is why standard Gibbs free energies can be used as tabulated inputs in the ΔG calculation.

Why does the derivation rely on a logarithmic pressure ratio?

For ideal gases, the lecture uses an expression where the change in free energy depends on the natural logarithm of pressure (or equivalently concentration) terms. This produces a term like ln(p_products/p_reactants). Because equilibrium constants are defined from ratios of product and reactant activities (approximated by pressure or concentration ratios for ideal behavior), the logarithmic ratio becomes the bridge to K.

How does the lecture transform ΔG into a form involving equilibrium constants?

It writes ΔG as (sum of standard Gibbs free energies of products) minus (sum of standard Gibbs free energies of reactants), plus a term involving pressures of products and reactants. Then it applies the equilibrium condition ΔG = 0. With ΔG set to zero, the remaining pressure-ratio factor is identified with the equilibrium constant, producing the final relationship between ΔG and K at the reaction temperature.

What does the final relationship let chemists do with K and ΔG?

It allows one to compute Gibbs free energy change for a reaction when the equilibrium constant is known at a given temperature (and under the pressure/ideal-gas assumptions used). Conversely, if ΔG is known, the equilibrium constant can be inferred. The lecture emphasizes that Gibbs free energy depends on temperature and pressure, so the relationship is tied to those conditions.

Review Questions

At equilibrium, what value must ΔG take, and how does that condition affect the derived ΔG expression?
In what way does standard Gibbs free energy differ from ordinary free energy, and why does that matter for calculations?
How does the logarithmic dependence on pressure (or concentration) lead to an expression involving the equilibrium constant?

Key Points

1
At equilibrium, the Gibbs free energy change satisfies ΔG = 0.
2
ΔG sign determines reaction direction: negative means spontaneous, positive means non-spontaneous.
3
Standard Gibbs free energy is defined at specified standard temperature and pressure, unlike free energy which varies with actual conditions.
4
For ideal gases, ΔG includes a natural-log term of pressure (or concentration) ratios of products to reactants.
5
Applying ΔG = 0 converts the pressure-ratio term into the equilibrium constant, linking thermodynamics to equilibrium composition.
6
With the equilibrium constant at a given temperature (and under the assumptions used), Gibbs free energy change for the reaction can be determined.

Highlights

The equilibrium condition is the thermodynamic trigger: setting ΔG = 0 turns the pressure-ratio term into the equilibrium constant.

Standard Gibbs free energy is comparable across labs because it is measured at fixed standard temperature and pressure.

The derivation uses an ideal-gas logarithmic dependence on pressure to connect ΔG with K.

Topics

Gibbs Free Energy
Chemical Equilibrium
Equilibrium Constant
Standard Gibbs Free Energy
Ideal Gases