# 2.6: Gibbs Free Energy

Most of the time, ATP is the “storage battery” of cells (See also ‘Molecular Battery Backups for Muscles below). In order to understand how energy is captured, we must first understand Gibbs free energy and in doing so, we begin to see the role of energy in determining the directions chemical reactions take. Wikipedia defines Gibbs free energy as “a thermodynamic potential that measures the "useful" or process-initiating work obtainable from an isothermal, isobaric thermodynamic system,” and further points out that it is “the maximum amount of non-expansion work that can be extracted from a closed system; this

maximum can be attained only in a completely reversible process.”

Mathematically, the Gibbs free energy is given as

\[G = H – TS\]

where H is the enthalpy, T is the temperature in Kelvin, and S is the entropy.

At standard temperature and pressure, every system seeks to achieve a minimum of free energy. Thus, increasing entropy will reduce Gibbs free energy. Similarly, if excess heat is available (reducing the enthalpy), the free energy can also be reduced. Cells must work within the laws of thermodynamics, as noted, so all of their biochemical reactions, too, have limitations. Now we shall consider energy in the cell. The change in Gibbs free energy (ΔG) for a reaction is crucial, for it, and it alone, determines whether or not a reaction goes forward.

\[ΔG = ΔH – TΔS,\]

There are three cases

- ΔG < 0 - the reaction proceeds as written
- ΔG = 0 - the reaction is at equilibrium
- ΔG > 0 - the reaction runs in reverse

For a reaction aA <=> bB (where ‘a’ and ‘b’ are integers and A and B are molecules) at pH 7, ΔG can be determined by the following equation,

\[ΔG = ΔG°’ + RT\ln \dfrac{([B]^b}{[A]^a}\]

For multiple substrate reactions, such as aA + cC <=> bB + dD

\[ΔG = ΔG°’ + RT\ln \dfrac{[B]^b[D]^d}{[A]^a[C]^c}\]

The ΔG°’ term is called the change in Standard Gibbs Free energy, which is the change in energy that occurs when all of the products and reactants are at standard conditions and the pH is 7.0. It is a constant for a given reaction.

In simple terms, if we collect all of the terms of the numerator together and call them {Products} and all of the terms of the denominator together and call them {Reactants},

\[ΔG = ΔG°’ + RT \ln \dfrac{Products}{Reactants}\]

For most biological systems, the temperature, T, is a constant for a given reaction. Since ΔG°’ is also a constant for a given reaction, the ΔG is changed almost exclusively as the ratio of {Products}/ {Reactants} changes. If one starts out at standard conditions, where everything except protons is at 1M, the RTln({Products}/{Reactants}) term is zero, so the ΔG°’ term determines the direction the reaction will take. This is why people say that a negative ΔG°’ indicates an energetically favorable reaction, whereas a positive ΔG°’ corresponds to an unfavorable one.

Increasing the ratio of {Products}/{Reactants} causes the value of the natural log (ln) term to become more positive (less negative), thus making the value of ΔG more positive. Conversely, as the ratio of {Products}/{Reactants} decreases, the value of the natural log term becomes less positive (more negative), thus making the value of ΔG more negative.

Intuitively, this makes sense and is consistent with **Le Châtelier's principle** – a system responds to stress by acting to alleviate the stress. If we examine the ΔG for a reaction in a closed system, we see that it will always move to a value of zero (equilibrium), no matter whether it starts with a positive or negative value.

Another type of free energy available to cells is that generated by electrical potential. For example, mitochondria and chloroplasts partly use Coulombic energy (based on charge) from a proton gradient across their membranes to provide the necessary energy for the synthesis of ATP. Similar energies drive the transmission of nerve signals (differential distribution of sodium and potassium) and the movement of some molecules in secondary active transport processes across membranes (e.g., H+ differential driving the movement of lactose). From the Gibbs free energy change equation,

\[ΔG = ΔH – TΔS\]

it should be noted that an increase in entropy will help contribute to a decrease in ΔG. This happens, for example when a large molecule is being broken into smaller pieces or when the rearrangement of a molecule increases the disorder of molecules around it. The latter situation arises in the hydrophobic effect, which helps drive the folding of proteins.

### Contributors

Dr. Kevin Ahern and Dr. Indira Rajagopal (Oregon State University)