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Gibbs free energy (2013)

Page history last edited by Joe Redish 8 years, 1 month ago

Working Content > Thermodynamics and statistical physics

 

Prerequisites:

 

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. This means that if the entropy of the system we are considering decreases, 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  increases by at least as much as the energy of our system is decreasing.

 

As a result, if our goal is a transformation of energy, 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 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! 

 

Exactly what free energy we use depends on what system we are describing. Since most biological systems function at a constant pressure and temperature, 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:  i.e., 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) 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 evolve in the direction of decreasing G.  

 

To understand why this expression for G makes sense, let's look at each part of the equation individually:

 

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 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 it's not 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.

 

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 1/11/12, intro and edits by Joe Redish 2/3/12

Wolfgang Losert 2/5/2013

 

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