Brief Review of Thermodynamics

I know, I know, yack! But we need to have a brief review of physical chemistry, thermodynamics in particular, if we are to understand many of the porcesses that occur with biological systems and biomolecules. For example, we have mentioned last time that one of the characteristics of living systems is that they fight equilibrium, and that to do that they have to do work, i.e., burn energy. Every time we have energy conversions involved, we will have to deal with thermidynamics.

System and Surroundings

The first things we have to define when we talk about anything in thermodynamics (or any science, for that matter), are the system and the surroundings.

The system is the part of the universe that we are intrerested in analyzing or studying. It can be as simple as a single molecule, or as complex as a whole petroleum refinery plant, or even more complex, a human being.

The surroundings is, as its name implies, everything else. If our system is a molecule, the surrounding will be the solvent in which it is dissolved, other molecules around it, the flask in which the mixture is contained, the lab around it, the building around the lab, and so on.

The combination of the system and the surroundings make, intuitively, the Universe (yep, the whole Universe). We will see that most things in thermodynamics happen when heat or work is exchanged between the system and the surroundings. Having defined this concepts, lets review the Laws of Thermodynamics.

There are two type of systems. When we refer to chemical reactions with a defined number of moles of reactants as the system, we are talking about a closed system. There is no mass coming in or getting out of the system. If left alone, these systems will almost always reach equilibrium (as we will see below). Thus, a closed system is not a good thing to be if you are a living organism.

Living organisms are open systems. That means, mass can get in or out of the system. This means that a reaction can be kept going for as long as, say, we keep adding fuel to it. In a analogous way, if your car was a closed system it would have a usefulness of about 300 miles. Now, since you can put gas into the tank (open system), your car will go for 200 thousand miles, and theoretically, for more than that.

First Law of Thermodynamics

The postulate of the first law is very simple: The energy of the Universe is constant. Another way of postulating this is that no energy is created or destroyed; Energy can only be transformed. As we mentioned above, the only thing that can happen to the energy is to go from the system to the surroundings or vice versa. Since everything is relative, we can focus on the system. We can therefore see the energy change of the system after a particular event (chemical or otherwise). The change in energy of the system, DU, is defined as the difference between the heat abosrbed by the system from the surroundings (q) and the work generated by the system on the surroundings (w):

Both heat and work are measured using the same units, Kcals (or Kcals/mole, depending on the system). However, they are very different from one another. Heat is originated from the random collisions between molecules (those of the system and those of the surroundings), and work is defined as a the product (actually, path integral) of a force and the distance covered by the entity exherting this force. Thus, when we heat the system, we are actually making more collisions between molecules take place. When the system does work, it actually moves something. In biological systems, this can be a membrane expanding under constant pressure, or the movement of a fibril in fluids.

Most work in biochemical systems is done under constant pressure. Since we are at constant pressure, the work involved in the expanssion of a gas (which as we will see if the type of work we are interested in) can be easily calculated as P * DV. At this point is useful to introduce another concept from thermodynamics, the enthalpy. We define ehthalpy as

The enthalpy can be viewed as the sum of work and energy in a system. It is now easy to calculate changes in enthalpy after a certain process in a system. Since we are working at constant pressure, we get:

Note that since we are working at constant pressure, we redifined the heat as q_p. Now, since we are considering that the only work is of the P * DV type, we can rewrite our enthalpy expression:

We see that the heat at constant pressure equals the enthalpy. Now we can rewrite the expression for the energy change:

Since most volume changes in biological systems are very very small, the term P * DV becomes insignificant, and the energy change, DU, can be calculated from the change in enthalpy, DH, which we know how to obtain easily for chemical reactions.

The size and sign of enthalpy of a process can tell us a lot about the process. The larger the DH, the more energy is being transferred (duh). Now, if DH is negative, we say that the process is exothermic, and that heat is realesed from the system to the surroundings. A simple example is the dilution of an acid or base with water. When DH is positive, we say that the reaction is endothermic, and heat is being transferred from the surrounding to the system (the system absobrs enegy). Again, a simple chemical reaction that reflects this is the disolution of a salt in water (the container gets cold...).

Now, enthalpy alone, and therefore the first Law of Thermodynamics, is not enough to tell us what we are really interested in: If a process is spontaneous or not. Imagine two containers, one cold and the other one hot that are placed in close contact. Intuitively, we know that the colder container will get warmer and vice versa. However, the first law of Thermodynamics just says that either process is allowed, as long as the total energy is constant. We need something else to determine spontaneity...

Second Law of Thermodynamics

To solve this problem we have to look at the second Law of Thermodynamics, which says that spontaneous processes are those in which we gain more disorder at the end. The other way of putting the second Law, which you probably have heard, is that the disorder of the Universe is always growing. How can we picture this? First, lets define W as the number of possible energetically equivalent arrangements of a system. This can be viewed as a measurement of dissorder: The larger W is, the more ways we can have the system (with the same energy) arranged, and therefore the more disordered. Now, lets consider a system in which two containers, one empty and the other full of a gas, with a total of N molecules, are connected, and the valve opened:

Since we have N molecules and 2 bulbs, the number of possible distributions of molecules between the two flasks will be 2^N. Since we have N molecules, the number of energetically equivalent arrangements, or states, will be (N + 1): We can have 0, 1, 2, ..., N - 1, or N molecules in the left flask. We can also calculate the probability of placing L of the N molecules in the left flask. The number of different ways of doing this , W_L, will be:

And the probability of a state with L of the N molecules in the left flask is

If you did all the math involved, you would find out that the maximum probablity is obtained when L = N / 2, that is, when both containers are equally filled up. That is because that state is the one in which we have the largest number of different ways of arranging the molecules in the system. As the systems get bigger (numbers close the 10²³), you would start seeing that the probability of having any other state other than half and half would be so small that can be considered zero.

Thus, you can clearly (?) see that the spontaneous process (both flaskes getting half-filled) is that one in which we have the largest number of arrangements with the same energy (the largest W).

Since the number of ways of arranging the molecules of a real system, which are in figures in the order of 10²³, is huge, things get very large and cumbersome to work with. Instead of using this, we use a quantity defined as the entropy of the system, which is:

As you know, the logarithm is a great way of making things smaller. k_B is known as the Boltzmann constant. If we go back to our two bulb system, and analyze it a bit, we would see that for the case in which W is higher (half and half), S = N k_B ln( 2 ), which is the maximum entropy value. For the case in which all molecules are in one side ( W = 1), S = 0, so this won't happen. What happens after the molecules equaly distribute? We will still have movement back and forth, but the whole thing looks static, at least macroscopically. We call this state equilibrium.

As we saw above, for a constant energy process ( DU = 0 ) to be spontaneous, DS > 0. If we generalize, we can say that any spontaneous process causes the entropy of the Universe to grow:

A system that reaches equilibrium most likely has reached its state of maximum entropy. This is very important, because it rules all biological processes, and, as we will see later, explains why we need to eat. We will see that the only way of moving away from equililbrium, and therefore ordering a system, is to use up energy from the surroundings. This in turn means increasing the entropy of the surroundings.

Measurement of DS

Now, is it possible to meassure entropy in a biological system by calculating the W ways of arranging a system? Not a chance. Fortunately, very smart fellows have come in our aid. An easier and more useful definition of change of entropy in a system is related to the heat involved in the process and the temperature at which it occurs. DS can be defined as:

These two quantities are easily masured in any system. Unfortunately, this gives us only an idea: DS will be equal or larger than q / T. The only case in which the values are equal is when the process is reversible, which no real process is.

Free Energy

So, now we have a way of estimating the DS of a certain process. However, a process for a certain system can still be spontaneous even when DS < 0. Think of forming water from oxygen and hydrogen. This is a process that is spontaneous, however, it has a negative  value (we go from 2 molecules to one, we order things). This is because we haven't considered the DH of the system, which for the case of water formation is large and negative (exothermic).

The real parameter that indicates spontaneity is the DS of the Universe for that process, which is pretty hard to measure. Therefore, we need something else. This something else is the Gibbs free energy of the process, G, which relates the S and the H of a process:

For systems that only do P * DV type work, and for constant T and P, we get that the change in free energy of a process is:

Since in a spontaneous process DS s q / T, DGc 0 is the criteria used to define spontaneity in this type of system. Spontaneous processes, those with DGc 0, can be used to generate work without adding energy, and are called exergonic. Processes in which DG s0 need energy to proceed, and are called endorgonic. If for a process we arrive to a DG = 0, the forward process and the reverse process are balanced, and we say the process is in equilibrium.

This last remark is pretty handy. As you probably recall, the free energy of a reaction of the type:

Can be described as:

Where DG^o is the standard free energy (the one when all the solutions are species are in 1M concentration at normal temperature and pressure). The term inside the logarithm is the equilibrium constant. As we saw before, a system is in equilibrium when DG = 0, so

This means that from thermodynamic measurements of the free energy of a reaction we can obtain equilibrium constants, and therefore concentrations for a certain reaction, or vice versa.

The thermodynamic concepts that we discussed today are very important in every aspect of biochemistry. For example, by looking at the last equations relating free energy with equilibrium, we can see how to maintain a process continously in the 'spontaneous' mode. We basically have to either add or remove one of the components of the reaction in a manner such that the is negative. As we will see later in the course, this is what all living organsms do: By changing concentrations of reactants and products, processes that should not happen (or should go in the reverse direction) are made spontaneous.

As you probably gathered by now, life is the capacity of fighting disorder and equilibrium...