An analogy for Gibbs free energy

So in this video I want to go ahead and talk more about the mathematical expression for delta G, or the change in Gibbs free energy, which I've went ahead and already written out here. So let's remind ourselves that this change in free energy is a quantity that is equal to the change in enthalpy, or heat content for a reaction minus the temperature at which the reaction is run times the change in entropy or, broadly speaking, disorder in going from reactants to products for a particular chemical reaction. OK so that was kind of a mouthful, right? So my ultimate goal in making this video is to ultimately present to you an analogy that I hope will give you a better intuitive feel for how the terms of enthalpy and entropy contribute to the overall sign of delta G. Because remember it is the sign of delta G, that is whether it's negative or positive, that tells us whether or not a reaction is spontaneous or nonspontaneous. But before talking about analogies let's go ahead and get a better handle on enthalpy and entropy from more of a chemistry and mathematical lens. So recall that enthalpy is really a good proxy for the change in bond energy that occurs during a reaction, that is to say whether or not energy was released or absorbed during a bond rearrangement. And because all systems want to achieve their lowest possible energy, if we're talking about enthalpy in isolation from entropy, let's say, generally speaking we might intuitively say that having a negative delta H, which describes a release of energy-- remember-- going from reactants to products, would be more favorable than a positive delta H value. On the other hand, the second law of thermodynamics says that all systems tend toward disorder and since entropy, or the chain in entropy, is a measure of whether something is getting more disordered or less disordered. In isolation, if we're just talking about entropy, we might intuitively say that having a positive delta S value, that is to say describing an increase in entropy from going to products from reactants, would be more favorable than a negative delta S value. Now mathematically our equation supports our hypothesis, right? A negative number minus a positive number will always be negative. That is to say regardless of what the temperature is, the delta G value will always be negative, the reaction will always be spontaneous. But since delta H is not always negative, and since delta S is not always positive for all reactions, what will the delta G be for other combinations of delta H and delta S? So to explore this further I thought we could go ahead and create a table like such to write out essentially the different signs that delta H and delta S could take on, and determine for each of these combinations whether the delta G will be positive or negative. So we've already gone through one. We've said that when delta H is negative, and delta S is positive, delta G will be negative. But of course there are other combinations too, delta H could be positive, and delta S could be negative, or both delta H and delta S could be positive, or delta H and delta S could both be negative. So let's go ahead and take these signs and plug and chug into our equation above. So if we have a positive number and we're subtracting a negative number, we are always going to get a positive number. So for all situations in which delta H is positive and delta S is negative we will have a positive delta G value. This makes sense, right? Because it's essentially the opposite of the previous situation. But what about the case in which you have a positive number minus another positive number? Well, you might say it really depends on the relative magnitude of these two numbers, right? Because if you have a big positive number minus a smaller positive number you would get a positive number. But if you had a small positive number minus a bigger positive number you would get a negative number. So ultimately it really depends, delta G can either be positive or negative. And this value of temperature plays a pretty big role in kind of tipping the scale towards one direction or another. Because if it's really high, for example, our second term might be more positive, and therefore this overall value might be more negative. But if it's not very high and delta H is really, really large, well, perhaps then delta G would be positive. So finally let's look at the situation where we have a negative number-- in our last row-- minus another negative number. Well, you might say again this really depends, right? If you have a small negative number and you're adding a very large positive quantity to it, it could be positive. But if it's a smaller number that you're adding on to a negative number, it could still be negative. So again this really depends, and again temperature plays a big role in tipping the scale one direction or the other. Now for the analogy that I want to present, I really want to focus on these last two rows here in which temperature plays a big role in determining whether or not a reaction is spontaneous or not. So how can we understand this trade-off between delta H and delta S at high or low temperatures? So I'm going to go ahead and scroll down and I'm going to present an analogy that I hope isn't too corny. But hopefully it's useful, and you might even consider thinking of your own analogy to help you understand this trade-off between entropy and enthalpy. So imagine you are a chemistry student, and for spring break you decide to go with your friends somewhere really tropical, where there are a bunch of palm trees and great beaches. And one day you and your friends go ahead out to one of the beaches and you and a couple of friends set out a large beach towel and decide to sunbathe on the beach, while some of your other friends decide to go out into the ocean and swim. Now of course being a chemistry nerd you describe this phenomenon in terms of a chemical reaction, that is to say a reaction in going from lying down on a beach towel to swimming in the water, or the reverse that is going from the water to lying on a beach towel. Now at this point you begin thinking back to your equation for delta G, which remember is equal to delta H minus the temperature times delta S. And you begin to realize that the forward reaction that is going from a beach towel to the ocean is more favorable, that is to say if more of your friends are in the water when the sun is really at its peak, and everyone really wants to go to the water to cool off. On the other hand, the reverse reaction that is going from the water to the beach towel is more favorable when the sun is less intense, when it's less scorching hot outside and it's easier to relax on the beach. In other words, you conclude the sun is figuratively, or perhaps even literally here, analogous to the temperature variable in our equation. Depending on how hot the sun is ultimately determines whether or not it is more spontaneous to go into the water or come out of the water. Now since you happen to also be an English major and love metaphors, you go ahead and think up a metaphor for both the enthalpy and entropy contributions in your hypothetical chemical reaction. Enthalpy, you decide, you can think about in terms of the amount of energy that you're expending whether you're on a beach towel or in the water. So we all know that lying on a beach towel is super relaxing and very peaceful, and requires little to no energy and you're in a very, very stable place. But on the other hand, when you're swimming in the water it requires a lot of energy be expended because you're of course swimming and throwing around a beach ball. On the other hand-- I'm going to go ahead and scroll down here-- lying on a beach towel can kind of get boring after a while, right? But being in the water is lots of fun, and there's really never a boring moment. And this interplay between fun and boring is something that you can kind of think of as your entropy variable. Now in an ideal world you'd want to be both relaxed and having fun. And in chemistry language that's really another way of saying that you'd want to have a negative delta H value and a positive delta S value. But of course in this particular setup, these things are mutually exclusive. So what wins? Ultimately, in the end, you conclude that when the sun is out this fun term, or what we're kind of calling entropy wins, and the forward reaction is spontaneous. But when the sun isn't as hot outside this relaxation, or enthalpy term, wins and the reverse reaction is spontaneous. And with this mighty conclusion that you have come up with on this relaxing day at the beach you go home and reminisce about how chemistry and life are far more similar than you once thought.