Entropy and free energy

If you’ve gotten this far in my little thermodynamics fun-fest, you should be familiar with the terminology, the basic ideas, and the many joys of enthalpy.  If you’re not familiar with these things, then I’d strongly recommend that you go familiarize yourself with them by looking at those tutorials.

In any case, it turns out that thermodynamics is more than enthalpy.  Though the whole exothermic-endothermic thing plays a big role in chemical reactions, it’s not the only thing that’s going on.  Let’s have a look at some of the other big players in the thermodynamic world.

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By the way, you’ll see the molecules better if you have one freakishly huge eye.


The random world of entropy:

My Uncle Carl passes gas after Thanksgiving dinner.  Now, that wouldn’t be so bad if the gas would just stay near his… behind… but it tends to wander all over the room.  Whether Carl is sitting next to you (God help you) or on the couch downstairs, his stench will find you.

The big question:  Why will the stench find you?  I mean, if Carl is sitting across the room, why don’t the molecules of his former meal just stay near him?  Shouldn’t they be a local phenomenon rather than a household or neighborhood event?

dog

More importantly, why do we let him blame it on the dog?

One of the reasons for this is entropy (S).  It turns out that there’s a driving force in the universe for stuff to get less ordered over time, and the movement of Carl’s effluvia from a small area to a big area makes it more disorderly.  In the same vein, if you were to open the door on a space capsule, the air will rush out because moving it over a large area is more disorderly than keeping it in one place.

As with enthalpy (H), entropy isn’t something that has an absolute value that you can write down.  After all, when you make something happen, it’s impossible to say how random things were both before and after the change, but you can certainly measure how much more or less random they got.  It’s like when you climb a ladder:  You may not know how far above sea level you are (i.e. your absolute altitude), but when you get to the top rung you know that you’ve travelled two meters upwards.  Your absolute altitude doesn’t matter – it’s the change in altitude.  As a result, we always talk about the change in entropy for a process as the term ΔS.

Though you may not think this causes all that big an effect, it turns out that it does.  In fact, the second law of thermodynamics phrases it like this:  The entropy of the universe is always positive for all spontaneous processes.

This means that unlike enthalpy, which stays constant, the entropy of the universe is always getting bigger.  The second law says that you can make a particular system more ordered (i.e. compressing a gas or freezing a liquid), but doing so will create more randomness in the surroundings than was removed by the process itself.  Put another way: If you were to freeze some water in your freezer (a process that has negative entropy), the refrigerator itself must have become more random through its workings than the amount of entropy that was taken from the water.


Fun fact:  There is an argument against evolution in which it’s said that evolution can’t be possible according to the second law because it results in a more ordered system.  This is true – evolved things are less random than the things they evolved from – but ignores the vast increase in randomness caused by the workings of the sun to generate the energy that drives the process.  While the system (organisms) get less random, the universe as a whole becomes more random.

Monkey

After escaping from the zoo, Mr. Bubbles took his friends to the woods and started their own society. And waited for the right moment…


How to calculate entropy

This is either simple or difficult, depending on how specific you want to get.  I’ll start with simple.

The simple way:  Use your intuition

If somebody asks you whether a system undergoes an increase or decrease in entropy, see if you can figure it out using common sense.  For example, let’s consider some phase changes:

  • When a liquid boils, the molecules go from being bound to one another with intermolecular forces to flying all over the place at high speed.  This should suggest that boiling is a process in which the system increases in entropy (and likewise, if you condense a gas, the entropy of the system will decrease).
  • When liquids freeze, the molecules go from wiggling around each other to being locked in some kind of lattice.  Because a lattice is considerably more organized than a liquid, the entropy of the system decreases during freezing (and likewise increases during melting).

Similarly, you can figure out a rough idea for entropy change for many reactions.  For example, when you undergo a synthesis reaction, more complex molecules are built from less complex ones (example:  H₂ + O₂ → H₂O₂).  Similarly, a decomposition reaction (such as 2 H₂O₂ → 2 H₂O + O₂ cause an increase in randomness because more molecules of product are present).

The more complicated way to find entropy:  Do the same stuff that you do for enthalpy

In the last tutorial I talked about the various ways of finding ΔH for a process using heats of formation and whatnot.  Entropy calculations are done in exactly the same way, so if you have the same starting information for entropy that you did for enthalpy, you should be in good shape.  In a very literal sense, you can find the ΔS for a process by using the methods in that tutorial and substituting the world “entropy” for “enthalpy” and the term “ΔS” for “ΔH”.  Easy peasy.


Free energy:  Will it happen?

Now that we’ve got a good grip on enthalpy and entropy, it’s time to use these things to predict whether or not a reaction will actually take place.  The term we’ll use to describe this is something called Gibbs free energy, and the change in free energy is known by the symbol ΔG.

Free energy is kind of a complicated idea, because if you get down to all of the details and math behind it, it involves a lot of stuff we don’t want to get into.  However, if you simplify it considerably, free energy describes the ability of a system to do things.  As a result, if a system is less able to do things after a change, we’d say that the ΔG for this change was negative.  If the system is better able to do things after a change, ΔG is positive.

To put it another way, we can tell a lot about a change by looking at its ΔG value:

  • If ΔG is positive, the system is able to do more stuff than before the change.  Basically, it has more free energy, so the process will require energy to be dumped into it to occur and it will not be spontaneous.
  • If ΔG is negative, the system is less able to do stuff than before the change.  The system has less free energy, which tells us that the process dumped energy into its surroundings, which means that it is spontaneous.
  • If ΔG is zero, the system is at equilibrium.  Essentially, there’s no free energy difference either way, so the process will neither be spontaneous or nonspontaneous.  It will instead be at equilibrium, which you can read about here.

Fun fact:  If a process is not spontaneous, then the reverse of that process is spontaneous.

177px-Josiah_Willard_Gibbs_-from_MMS-

Josiah Willard Gibbs, the man for whom Gibbs free energy is named. He was also well-known for his “party down” attitude.


Do the math:  Finding ΔG

There are several ways to calculate ΔG.  The first way is to go back to the tutorial about enthalpy and substitute ΔH with ΔG (the same thing you did before with entropy). However, the most common way of doing this is using this equation:

ΔG = ΔH – TΔS

Take a minute to look at this equation and consider its awesomeness.  Great!  Now that you’re done pretending to think this is awesome, let’s break this down.  What this equation tells us is that a process will be more likely to be spontaneous (ΔG will be negative) given the following:

  • The process is exothermic (ΔH is negative).  Obviously, if ΔH is negative, this will make it more likely that ΔG is negative.
  • The process results in an increase in randomness (ΔS is positive).  Given that the second term puts a negative sign in front of TΔS, this tells us that a positive ΔS will result in a negative overall term.  However, what this equation also tells us is that temperature plays a big role in how important the ΔS term is, with high temperatures making entropy more important.

It’s also important to remember that T, in this case, is equal to the temperature in Kelvin. As result, if somebody asks you to do a calculation with this equation at a temperature of 25° C, remember to add 273 to get the temperature in Kelvin (298 in that example).

Aside from that, it’s all a matter of just plugging the numbers into equations to see whether something is spontaneous.

Example:  If the reaction A + B > C has a ΔH of -125 kJ/mol and a ΔS value of -14.2 J/K·mol at a temperature of 455 K, will it be spontaneous?

Answer:  Use ΔG =ΔH – TΔS to figure it out:

ΔG = (-125,000 J/mol) – (455 K)(-140.2 J/K·mol)

ΔG =  -63,000 J/mol or -63 kJ/mol

Because ΔG is negative, this process is spontaneous and the reverse process is not spontaneous.


Photo credits:

  • Big eye lady:  Image courtesy of marin at FreeDigitalPhotos.net
  • Farting dog:  Image courtesy of SOMMAI at FreeDigitalPhotos.net
  • Killer monkey:  Image courtesy of wandee007 at FreeDigitalPhotos.net
  • Party animal Gibbs:  Public domain via Wikimedia Commons.
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