Physical Chemistry, Part 27： The Helmholtz Free Energy

Hi, and welcome back to my videos for Physical Chemistry 1. For the past several videos, we've been talking about entropy and the ways that you can calculate it. Starting today, we'll have several videos that explore how entropy is connected to other thermodynamic properties of a system, like the enthalpy and energy. To begin, let's look at the energy. As we've seen several times before, it's easiest to study a system if we hold one or more properties of the system constant. This helps us to concentrate on the properties that we're the most interested in, like the energy or the entropy. Usually, the properties that we hold constant will be the temperature, volume, or pressure. So, to start, let's think about a system where it's the temperature and volume that are both constant. One thing that's important to notice here is that we're not assuming that our system is isolated. In other words, the system and the surroundings are in contact, so it's possible that energy might be entering or leaving the system. That's different than the situation we've had in all the previous videos, in which we've been looking at isolated systems. The main thing that we'll be interested in is whether or not a non- isolated chemical reaction will take place spontaneously. Let's start by thinking about the first law of thermodynamics. For the system we're looking at, in which the temperature and volume are both constant, what can we say about the variables in this equation? We know that the work is equal to the negative of the pressure times the change in volume. Since the volume is constant, that means the work is equal to zero, so we can drop it out of our equation. So, for this system, the change in energy is just equal to the heat exchange. Since we're trying to determine whether or not the reaction is spontaneous, it makes sense for us to also think about the entropy of the system. We know that the second law of thermodynamics tells us that the entropy is always greater than or equal to the heat divided by T. If we rearrange this equation a little, we have this. We can substitute this inequality into our expression for du to get this equation. So, this tells us that du is always less than or equal to Tds. It's important to keep in mind that this is true for a system where the volume and temperature are constant. With that in mind, let's remember that the two sides of the original inequality we used is equal for a process at equilibrium and greater than for a spontaneous process. In the same way, the less than inequality we got here is correct for a spontaneous process. Let's move the Tds to the left side of the equation. Now, we can group together the two terms this way. The reason we would do this is because written this way, the equation is telling us that for a spontaneous process, the change in u minus Ts will be less than zero, so it's a negative number. In other words, a process like a chemical reaction will be spontaneous if the change in u minus T times s is a negative number, and it'll be zero if the reaction is at equilibrium. This is such a useful result that the term in parentheses gets its own symbol and its own name. It's called the Helmholtz free energy in honor of the person who first recognized its importance, Hermann von Helmholtz, and it has the symbol capital A. Helmholtz made lots of contributions to the science of thermodynamics, but he was also very interested in the senses and how sensory perceptions are transmitted by the nerves. In order to study vision more deeply, he invented the ophthalmoscope, a device we still use to examine the interior of the eye. Anyway, as we saw earlier, the change in the Helmholtz free energy must be less than zero for a spontaneous process and equal to zero for a process at equilibrium. That also means that a reversible chemical reaction will start out with a negative value for delta A, and it will slowly increase until it reaches zero, at which point the reaction will be at equilibrium. Notice that if delta A is greater than zero, that doesn't mean the reaction is impossible, just that it isn't spontaneous. If we want such a reaction to occur, it will be necessary to add energy to the system. Also, notice that based on our definition of the Helmholtz free energy, delta A is equal to this. That means that delta U minus T delta S is less than zero for a spontaneous process. Since the temperature and entropy will both be positive numbers, that means that this term is positive. What does that tell us? Well, the main thing to realize for now is that this equation shows that if we raise the temperature, the left side of the inequality will decrease. In other words, it'll become more spontaneous. You've probably noticed that in the lab. Many chemical reactions that are slow or impossible at one temperature may become spontaneous when we increase the temperature. So, the Helmholtz free energy gives us some useful information about the spontaneity of a chemical reaction. But it has one drawback. It's meant to be used for systems where the volume and temperature are both constant. Unfortunately, constant volume processes are fairly rare. That's the situation we have when we do a bomb calorimetry experiment, but other than that, we don't see constant volume processes very often. Instead, it would be more helpful to have a property we could use at constant pressure, which is a very common situation. And there is such a property. That's what we'll look at in our next video. We'll find out that the property we can use at constant pressure is one that we haven't discussed yet. And once we learn about it, we'll have the last of the major thermodynamic properties that you'll want to know about. I hope you'll join me for that. But until next time, have a good week!