Motivating Free Energy

Biological systems are continually occupied with collecting, organizing, storing and utilizing energy.  Systems from cells to organs to organisms collect energy in various forms, control the flow of materials to and from compartments within cell, organs, and organisms, and maintain their temperatures to prevent harmful side effects such as the unraveling of proteins (denaturation) and the failure of cell membranes. From a biology perspective, metabolic pathways provide a wonderfully complex roadmap to many of these processes and highlight how energy flows through a biological system. 

The key idea is this: energy tends to spontaneously flow to increase entropy — basically to spread the fluctuating energy out more evenly. If the system is at equilibrium, there is equal flow of energy in every direction and you can't do much with it. So a biological system wants to find places where energy is localized and extract work from it for the processes of life and it uniformizes.

Here we consider the flow of energy from a physics perspective, and ask how the  foothold principles of physics and thermodynamics we already studied constrain how energy can flow in a biological system and how much energy is available to do biological work. We will tackle the following questions:

• Where is localized energy available?
• What conditions will allow that localized energy to spontaneously transform?
• How much of that energy can I use?

Seeing when work can be extracted in a toy model

As is usual in physics, we gain insight by consider as simple an example as we can construct: a box with gas on one side and vacuum on the other as shown in the figure below.

If we remove the dividing membrane and allow the molecules of the gas to move into double its original volume, the molecules will hardly notice it. Whatever kinetic energy they have originally they will still have. This tells us that their temperature will not change, but since their density will drop by a factor of 2, the pressure will drop by the same factor as the volume increases. The molecules will have the same average speed, but will collide with the walls less often.

When the molecules were all on the left, we might have extracted some energy from the gas. For example, instead of simply removing the partition, we might have punctured a small hole in it and let the gas stream out from the left into the right. We could then have put a little wind turbine in the path of the stream of gas and let it turn the turbine charging a battery. We would have extracted some energy from the gas — and the resulting temperature would be lower

But if we start when the gas is equally distributed on both sides, the fan would be hit by the same number of molecules from each side so it would not turn. So despite having the same amount of energy, when the gas is all on the left we can extract useful work from it. The it is spread out to both halves, we cannot.

The question we are interested in is, how much energy can we extract from the gas? All of it? Only some of it? We identify any energy that we can extract from system as free energy. To figure out how much this is we have to think about the constraints of the first and second laws of thermodynamics.

When we try to predict how a system will evolve — that it, how its energy and concentrations will spontaneously rearrange themselves, the first and second law of thermodynamics prescribe two tendencies:

• If any work is done by the system, we expect the system to move toward lower energy.
• From considerations of what we know about statistics and probability, as expressed by the Second Law of Thermodynamics, we expect the system moves toward higher entropy.

"Free energy" is a way of combining both of these tendencies into a single quantity that tells you how a system will tend to evolve. 

Constructing a free energy

There are different kinds of free energy applicable in different situations. To see how and why we construct them is a bit beyond the level of thermodynamics we are going into in this class. (You will need to do it when you study Physical Chemistry.) But we can develop a qualitative story that gives us some insight. We want to construct thermodynamic state functions — functions of the thermodynamic state ($P$, $V$, $T$, ...) that are well defined when the system (or a part of the system) is in thermodynamic equilibrium.

The quest for a free energy begins with the study of heat engines: machines that use a flow of thermal energy (heat) from a higher temperature ($T_H$) to a lower temperature ($T_C$) in order to extract work ($W$). While the 1^st law (energy conservation) might suggest that we could convert it all, the 2nd law (entropy of the universe increases) shows that if a certain amount of thermal energy, $\Delta U$, us transferred out of a hot reservoir, not all of it can be converted to work. A certain minimum amount of heat, $Q$, must be dumped in the cold reservoir. (See the Thermal Efficiency page in Wikipedia for details.)

This suggests that we might want to consider the internal energy minus the heat as a free energy function, $U - Q$. But this doesn't work because $Q$ is not a well defined thermodynamic state function. It depends on the connection between two thermodynamic states. But for a heat transfer $Q$, the entropy change is $\Delta S = Q/T$ and both $S$ and $T$ are well defined thermodynamic functions. Since $Q = T\Delta S$, we might try to define a free energy

$$F = U - TS$$

This works. $F$ is referred to as the Helmholtz free energy and is a well-defined thermodynamic state function. If it particularly useful in situations where the temperature and volume of the transforming system are constant.

But the free energy that is most useful in biology is the Gibbs free energy (named after Josiah Willard Gibbs), since it's defined for a system at constant temperature and pressure, which describes the environment inside a cell. We'll look at it in more detail in the follow on pages. But if our system is staying at constant temperature and pressure as it interacts with its surroundings via heat and work, the Gibbs free energy helps us sort out what will happen to the energy of the system.

Ben Dreyfus, Joe Redish, and Wolfgang Losert 2/5/2013, 4/11/19