# Gibbs free energy

#### Prerequisites

- Motivating Free Energy
- Example: Entropy and heat flow
- Enthalpy
- Example: Free energy of an expanding gas

## Transformations of a biological system must satisfy the 1^{st} and 2^{nd} laws of thermodynamics

In biology, chemistry, and physics, processes where energy changes from one form into another or from one location to another are of particular importance. These processes involve a transformation within the system of interest — a chemical reaction, a change in pressure or volume. These processes are constrained by the first and second laws of thermodynamics: the total energy of the universe is conserved, and the total entropy of the universe must increase.

In biological systems, we typically are interested in using an energy flow to *increase *the order of our system — arranging the parts of the system in the way that it performs a particular needed function. The result is that the entropy of our system will *decrease*.

In order for this to happen, to satisfy the 2^{nd} law, at least the same amount of entropy (and possibly more) must be "dumped" into the external world. (And at the same time the energy of the world external to our system must change by exactly as much as the energy of our system is changing but in the opposite direction.)

As a result, if our goal is a transformation of energy to create order, we can't use all the energy in the transformation to do work — that would violate the second law: some of the energy has to be dumped outside and increase the entropy of the surrounding.

What remains after that energy has been given away is the *free energy. *The *sign* of the free energy tells us whether or not *enough* energy has indeed been dumped into the environment such that the overall entropy of the universe increases (as it must for a process to occur).

*If the sign of the free energy change is negative, enough energy was dumped into the environment and the process can happen spontaneously. If the sign of the free energy change is positive, not enough energy was dumped into the environment and the process cannot happen spontaneously! *

## Which free energy? (Hint: Gibbs)

Exactly what free energy we use depends on what system we are describing. Since most biological systems function at a constant pressure ($p$) and temperature ($T$), the appropriate form is called the ** Gibbs free energy**.

Using the simple model of a heat dump, we can now figure out how much energy we can't use. If our entropy decreases by $ΔS$, then we have to dump an amount of heat equal to

$$Q = TΔS$$

This has to be subtracted off the energy we have available from our transformation.

Because we're dealing with systems at constant pressure, the energy we're using here (and balancing with entropy) includes the work done to keep the system at constant pressure: that is, the *enthalpy*.

Putting this together, Gibbs free energy ($G$) is defined as * *

$$G=H - TS$$

But we only care about the **change** in Gibbs free energy, and so (remembering that we're also at constant temperature so there is no $\Delta T$) we get the expression you've seen in chemistry:

$$ΔG = ΔH - TΔS$$

The sign of $G$ is defined so that a system will tend to spontaneously evolve in the direction of **decreasing** $G$.

## Understanding $\Delta G$ in extreme cases

Gibbs free energy is most useful in studying chemical transformations, which we won't consider here. But to understand why this expression for $G$ makes sense from the physics point of view, let's look at special cases in which each part of the equation dominates.

First of all, look at the minus sign. This means there are two terms ($ΔH$ and $TΔS$) influencing $ΔG$ in opposite directions. Depending on which one is larger, $ΔG$ can be either positive or negative.

Now look at $ΔH$. It has the same sign as $ΔG$ in the equation, so a negative $ΔG$ (the direction in which things will proceed) is more likely if there is a negative $ΔH$. Remember that

$$ΔH = ΔU + pΔV$$

so a negative $ΔH$ can happen when we have decreasing internal energy (e.g. forming chemical bonds), and/or decreasing volume (taking up less space). This term represents the tendency of systems to move to lower energy.

Next, look at the $-TΔS$ term. It has a negative sign in front of it, so $ΔG$ is more likely to be negative when $ΔS$ is **positive**. In other words, systems are likely to proceed in the direction of increasing entropy. Since entropy has units of energy/temperature (J/K), this means that $TΔS$ has units of energy. (If it didn't, this equation would make no sense!)

What is the role of temperature? If the temperature is low, then the $TΔS$ term is small, and the $ΔH$ term (which doesn't explicitly depend on temperature) is what matters. In other words, things move towards lower energy and that's all there is. Think about the Energy Skate Park, with the skateboarder starting from rest (or very small initial velocity); she's just going to go downhill. It's just plain old classical mechanics; there's no random motion to worry about. But if the temperature is high, then the $TΔS$ term is large (relative to the magnitude of $ΔH$), so random thermal motion is going to have more influence. The system has enough thermal energy to access higher-energy states (and won't just "fall down" to the lowest state), and because of probability, it will move to higher entropy.

What if $ΔH$ is very small? Then $ΔG$ is basically just $-TΔS$, and only entropy matters in determining which direction a system will evolve. An example of such a process is diffusion (of particles that aren't interacting significantly). This process is entropy-driven, and will proceed in the direction of particles being more spread out. All the arrangements of particles have the same energy, so energy differences aren't significant in this process.

What if $ΔS$ is very small instead? Then $ΔG$ is basically just $ΔH$, and only energy matters. It's like single-particle mechanics, where we only need to worry about the energy of (or the forces on) a single object, rather than the different arrangements of particles.

But sometimes $ΔH$ and $TΔS$ have comparable magnitudes. Then we can no longer use either of the toy models we have used so far. Many real systems are neither objects in an "energy skate park" nor randomly moving objects. Those are the cases when the quantitative expression is most useful, since the result may not be qualitatively obvious. Note that the answer depends on temperature $T$. At low temperature, essentially everything resembles an energy skatepark. At high temperatures, motion is random. It is in between where most life occurs and where we need to use Gibbs free energy to analyze which of the terms win.

## Some dangerous bends

There are a three tricky points to remember about Gibbs free energy.

- Gibbs free energy is a state function. As a result, it is well defined for every system (or part of a system) that is in (local) thermodynamic equilibrium.
- It's only the sign of the
*change*in the GFE that tells us if a system will make a transition from an initial state to a proposed final state. The sign of $G$ itself is not indicative of anything in particular. - The change in the GFE is only indicative of spontaneity for certain conditions: constant temperature ($T$) and pressure ($p$).

A final note: The sign of $ΔG$* *tells us which direction a reaction (or other process) will proceed. But what impact does this have on the **rate** of the process? Answer: NONE! $ΔG$* *is totally unrelated to how fast things happen! There are some processes that have a negative $ΔG$ (and are therefore "spontaneous") and yet proceed incredibly slowly. This is why, in biological systems, we need enzymes (and other catalysts) to speed things up (without changing $ΔG$).

Ben Dreyfus, Joe Redish, Wolfgang Losert 11/12, 2/13, 4/16/19

#### Follow-on

Last Modified: May 24, 2019