Gibbs Free Energy and Spontaneous Reactions

 

It’s Quiz Time!  Close your notes, clear your desks, and answer the following question: What does the following graph depict?

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A friend of mine said: “it’s an exothermic reaction!”  And to that, I said, “Incorrect!”.

This reaction is, in fact, an exergonic reaction, which is a fancy way of saying that the reaction releases more overall energy than it absorbs.  Another way to describe this reaction is to call is spontaneous.  If you also said the graph was an exothermic reaction, implying that it releases more heat than it absorbs, you should know that heat and energy are not the same thing.  Heat is a form of energy, but it is not the only one.  Here’s what I mean:

The Free Energy Equation

$$\Delta G=\Delta H-T \Delta S$$

This is the equation for the Gibbs Free Energy (or just “free energy”) of a reaction.  Free energy is the energy stored in the bonds of a material/molecule.  In the above equation, the free energy is denoted by the \(G\) (for Gibbs).  We use the \(\Delta\) (delta) operator to describe the change in \(G\) between products and reactants of a chemical reaction.  You likely already know that \(\Delta G=G_{\text -G_{\text \), and the same idea will hold true for the other two uses of the \(\Delta\) operator in this equation.

Also, the graph of the exergonic/spontaneous reaction above reveals that the free energy of the products is less than the free energy of the reactants.  We can therefore say that for an exergonic (or spontaneous) reaction, \(\Delta G<0\).

Another interesting thing we know is that the universe always tends to carry out reactions from high energy to low energy, which is why decreasing the free energy of a system is called “spontaneous”.  It pleases the universe, and thus is a favorable reaction.

Let’s take another look at this equation: \(\Delta G=\Delta H-T \Delta S\), and start to think about the right-hand side of the equation.  What is this stuff: \(\Delta H\) and \(T \Delta S\)? Well, the easy bit is \(\Delta H\), which refers to the amount of heat transferred during the reaction.  When this other mystery term: \(T \Delta S\) is equal to zero, then we simply get \(\Delta G = \Delta H\), and every exergonic reaction would also be exothermic.

But here’s the reality of the situation: \(T \Delta S\) is almost never equal to zero.  \(T\) refers to the absolute temperature of the system, which by definition is never zero.  And \(\Delta S\) is the amount of entropy gained or lost during the reaction.  Note that when the entropy of a reaction increases, the reaction become more spontaneous.

If you’ve never heard an in-depth explanation about entropy, don’t panic—you don’t need one to solve problems using the equation for free energy. Normally, \(\Delta S\) values are simply given to you during a problem, or you can find them in tables.  But for your own good, I’ll include a brief introduction to entropy here.

Entropy—The Universe’s Friend

If you’ve ever learned about entropy before, you’ve probably heard about the “messy room” model of entropy—the idea that when things get messier or more disorganized, entropy increases.  And that’s fine; the messy room model is okay.  But a more technical (and useful) way to think about entropy is like this: entropy is a measure of the number of ways a system can be arranged. Sometimes we can count the number of ways something can be arranged—or the number of “microstates” it has.  Most of the time though, the number of microstates is way too many to count, especially when we think about how many molecules we have to take into consideration in an average system.  When a group of molecules can move around freely, or if it can take up more space, its entropy will increase, because its number of arrangements increases.

Think of it this way: ice is a fairly rigid block of water.  Molecules of frozen water don’t have very many microstates, because they are stuck next to each other, unable to move freely.  The most they can do is vibrate in place, according to their absolute temperature.  Molecules of liquid water, however, can slip past one another at will, and can therefore spread out into puddles.  The number of arrangements for the molecules increases when ice transforms to water, and therefore entropy increases.  Similarly, when water vaporizes, the molecules spread out even further, and can be arranged anywhere throughout the entire room.  Entropy dramatically increases.

Now, of course, we can to consider a new question: if liquid and gaseous water are so high in entropy, and high entropy increases the spontaneity of a reaction, why does any ice exist?

Of course, the equation for free energy: \(\Delta G=\Delta H-T \Delta S\), reminds us that spontaneity is about three things: the heat involved in a reaction, the entropy involved in the reaction, and the temperature at any given time.  The process by which ice melts into water requires heat, and is therefore not favored by \(\Delta H\).  But it is favored by \(\Delta S\), because it increases the entropy of the water molecules.  The overall tie-breaker is the temperature.  At high temperatures, \(\Delta S\) takes over as the dominant force of the free energy equation, and the reaction proceeds spontaneously.  However, at low temperatures, \(\Delta S\) is not enough to overcome the value of \(\Delta H\), and the process is not spontaneous.  We know this intuitively already: ice melts at high temperatures, but stays frozen at low temperatures.  Not exactly rocket science.

When you arrive to a testing situation, you might get values for \(\Delta H\), and \(\Delta S\), and then plug them into the equation for free energy.  It’s also likely that you find an equation that looks like this:

A reaction with a negative \(\Delta H\) and a negative \(\Delta S\) is…

a. Always spontaneous
b. Never spontaneous
c. Only spontaneous at high temperatures
d. Only spontaneous at low temperatures

The answer to this question is: d. Only spontaneous at low temperatures.  When  \(\Delta H\) and \(\Delta S\) are both positive or both negative, the spontaneity of the reaction is dependent upon temperature.  I really don’t recommend wasting memorization space over which sign corresponds to which type of spontaneity—just look carefully at the free energy equation to answer the question.  And if you get stuck, you can always remember water.  When ice melts, the process is favored by entropy but not by heat.  Therefore, ice melts at high temperature.  The multiple-choice question above is the exact opposite: the process is favored by heat, but not by entropy.  Therefore, it occurs at low temperatures.

What did we Learn?

There’s more to spontaneity than just heat output in a reaction.  If heat exchange was the only thing that dictated reactions, then ice would never melt!  We don’t commonly think about entropy and the free energy equation in our daily lives, but they are still very important.  Most of all, we should have an appreciation for the fact that everything is more complicated than it first appears—just because your high school teacher told you that exothermic reactions are favorable doesn’t mean that there’s nothing else at play.  As always, there are some interesting caveats that make our life on earth possible.