# Entropy

Learning Goals

You will learn a new causative factor called entropy and the second and third laws of thermodynamics. You will also learn how to calculate the entropy changes in the system and in the surroundings.

Synopsis

All of us instinctively know that energy seeks a minimum value. For example, at certain temperatures, solids do exist. If liquid water finds itself at -10 oC and one atmosphere pressure, the liquid will spontaneously change into ice (note the use of spontaneous) rather than remain as a liquid. The solid state is a lower energy state than the liquid state. Similarly if water vapor finds itself at 50 oC and one atmosphere pressure, it will spontaneously change into a liquid. The liquid state has a lower energy than the gas state. We also know that a hot object will spontaneously cool to the temperature of its surroundings. A cool object does not suddenly spontaneously become hotter than its surroundings. From our brief study of organic chemistry (Chapter 11), you know that molecules exist in different structural isomers. Consider ethane, which can exist in a form where the hydrogen atoms eclipse each other or in a form where the hydrogen atoms bisect the angles (staggered) formed by opposite H-C-H sets. Below is ethane in the staggered form showing it from a side view and looking down the C-C bond.

The staggered form is the preferred form because it is the lowest energy form. These and many other examples inform us that energy seeking a minimum can be a driving force for a process to occur.

But we also know that a gas will spontaneously expand to fill a container. A gas does not spontaneously contract to occupy the smallest volume that it can in a container. This process of spontaneously expanding is not proceeding in the direction of decreasing energy. There are many processes that occur at constant energy. So we need another driving force to account for such observed behavior.

All of us are also very familiar with the concept of disorder. If you have a deck of cards in a neat stack on a table and someone picks them up and throws them up so that they come to rest scattered all over the room, you know that the disorder of the system of cards has greatly increased. Where once you had order, now you have great disorder. My office, and probably your room, seems to seek the situation of highest disorder!

The gas spontaneously expanding to fill a container is certainly in the direction of increasing disorder. Further analysis would show us that for many processes that occur at constant energy, the spontaneous change is in the direction of increasing disorder. You may also remember that an isolated system is one that has no energy exchange with its surroundings and thus remains at constant energy. We thus have found our second driving force and we call the system disorder the entropy content. The symbol for entropy is S. These deliberations lead us to the second law of thermodynamics:

"The direction of a spontaneous change in an isolated system is in the direction of increasing entropy."

A mathematical form of the second law of thermodynamics is that, if T is constant, S > q/T or = q/T. The equals sign is used for a reversible change and the greater than sign is for an irreversible change. You may remember that a reversible change is one in which each of the infinite number of steps in going from state A to state B is an equilibrium step. For a process to be irreversible, only one of the infinite number of steps needs to not be an equilibrium step. Obviously real processes are irreversible. S is a state function, which you will remember is a function whose change in going from one state to another state does not depend upon the path of the process. Because the change in entropy, S, does not depend upon the path, then the change in entropy is the same whether the process is reversible or irreversible. Thus in changing from a reversible process to an irreversible process, it is not S that is different, but it is q that is different. Remember that heat and work are not state functions. Thus we can write

S = qrev/T and S > qirrev/T with qirrev > qrev

When you remember that in thermodynamics we call the system and its surroundings the universe, then we can state the second law in another form:

"The entropy of the universe is continually increasing."

To determine the entropy of a system, we need the third law of thermodynamics:

"The entropy of a pure, perfectly crystalline substance at zero Kelvin is zero."

All of the above words are very important. If the substance were not pure then the impurities would impart disorder, or entropy, to the substance. If the substance were not a perfect crystal then the crystalline impurities would impart disorder to the substance.

Now that we have a definition of zero entropy, the physical chemist can determine experimentally the entropy of a substance at any temperature. The temperature chosen is 25 oC. The chemist does have to take into account any phase changes that occur in going from zero Kelvin to 25 oC. The entropy of a phase change is calculated using the reversible form of the mathematical form of the second law of thermodynamics, S = q/T. T is the temperature of the phase change and q is the heat change associated with the phase change. If the phase change is the process of changing from a liquid to a gas, the S would be equal to Hvap/Tb. The entropy of substances is also determined at standard conditions that includes a pressure of 1 bar. Such a reference pressure is indicated with a superscript zero: So. [Some texts will use a superscript "theta" for the 1 bar designation since for many years the standard pressure was 1 atmosphere and the superscript o was used to indicate 1 atmosphere.]

Table 20.1 in your text lists the entropy of a number of substances at 25 oC and one bar and you can find many more values in the Handbook of Chemistry and Physics. Looking at these values and thinking about the molecular situations a bit, one can easily develop some generalities:

• For the same or very similar substances the entropy of the gaseous state is larger than that of the condensed phase.
• As the complexity of the molecule increases (more atoms, more ways in which the atoms can move relative to each other), the entropy increases.
• For ionic compounds, as the strength of the bond increases, the entropy decreases.
• When a pure solid or liquid dissolves in a solvent, the entropy usually increases.
• The entropy of a gas increases when its environment changes from being dissolved to being free.
We are now ready to calculate the entropy change in a system. Consider the reaction

CO(g) + 2 H2(g) --> CH3OH(l)

First let's think about the entropy change in the reaction. On the reactant side we have 3 moles of gases and on the product side we have 1 mole of a liquid. This reaction is changing from a more disordered state to a more ordered state so we would expect S (Sreactants - Sproducts) to be negative since Sreactants > Sproducts. Now let's calculate S from the table values for the individual entropies.

Sosystem = So(CH3OH(l)) - So(CO(g)) - 2 So(H2(g)) = (1 mol)268.8 J/Kmol - (1 mol)197.6 J/Kmol - (2 mol)130.7 J/Kmol = - 332.2 J/K, which is negative as we had predicted.

Remembering that we say that the "universe" for something being studied is the system and the surroundings, the total entropy change associated with some change is

Souniverse = Sosystem + Sosurroundings

So now we need to calculate the entropy change in the surroundings. The entropy of the surroundings is calculated using Sosurroundings = qqsurroundings/T and if the pressure is constant this is equal to -Hosystem/T. From Chapter 6 you know how to calculate the [delta]Hosystem for this reaction with the result that Hosystem = -128.14 kJ (note that it is kJ). We then have that Sosurroundings = 430. J/K

We then have that Souniverse = -332.2 J/K + 430. J/K = 98 J/K

Since the total entropy change is positive, the reaction is proceeding in the direction of increasing entropy and thus according to the second law of thermodynamics, the reaction will proceed spontaneously toward products. If we had relied totally on the entropy change of the system we would have made the wrong prediction. Table 20.22 gives you a summary of the guidelines to use to predict the spontaneous direction of processes.

Review Questions

1. Go to Screen 20.6 on your CD-ROM for further illumination on this topic.
2. Will the following reaction proceed spontaneously toward the reactants or toward the products

CH4(g) + 2 O2(g) --> CO2(g) + 2 H2O(g)

3. For which of the following will entropy increase?
• melting of a solid
• freezing of a liquid
• mixing of liquids
• rusting of iron

On-Line Activity Here is an interesting site on some further aspects of entropy:

Web Author: Dr. Leon L. Combs