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Unit 9: Equilibrium and Advanced Thermodynamics—Balance in Chemical Reactions

Section 2: Microstates, Macrostates, and Entropy

In 1868, Austrian physicist Ludwig Boltzmann (1844–1906) published the key research that explained spontaneity. Because the numbers of particles involved in chemical reactions are astronomically large, Boltzmann used statistics to describe the positions and energies of the particles. Thus, Boltzmann is the founder of statistical mechanics, an approach that revolutionized the study of thermodynamics.

Two-Bulb Apparatus

Figure 9-2. Two-Bulb Apparatus

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Two-Bulb Apparatus

Figure 9-2. Two-Bulb Apparatus

When the valve between the two bulbs opens, the gas particles distribute themselves evenly throughout because an even distribution is statistically most likely.

Rather than trying to follow every particle in a reaction at the same time, he just looked at the chances of where the particles were likely to be, and from that, thermodynamics could be explained. Boltzmann's equations showed how particles distribute themselves in predictable ways, despite each particle moving around randomly.

Consider the two-bulb apparatus shown in Figure 9-2. Two glass bulbs connect to each other through a valve. In the top portion of the image, the valve is closed, and the bulb on the left contains a quantity of purple gas; the bulb on the right is empty. Opening the valve allows the gas to travel into the other bulb until the amounts of gas are equal on each side; this process is spontaneous. The reverse process, where the gas gathers together on one side of the apparatus, is nonspontaneous. Another interesting thing about this process is that it is clearly spontaneous for the gas to move, but it doesn't involve a change in enthalpy!

A Theoretical Two-Bulb Apparatus Experiment

Figure 9-3. A Theoretical Two-Bulb Apparatus Experiment

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A Theoretical Two-Bulb Apparatus Experiment

Figure 9-3. A Theoretical Two-Bulb Apparatus Experiment

A theoretical two-bulb apparatus containing four gas particles. The atoms can be in one of 16 possible arrangements at any given time.

To keep things simple, let's think of a two-bulb apparatus that contains four atoms of gas free to travel between the two bulbs. (Figure 9-3) At any particular point in time, the atoms will be in one of 16 possible arrangements. Each one of these arrangements, such as having all the atoms on the left and none on the right, or having the red, yellow, and blue atoms on the left and the green on the right, is called a "microstate." The microstate is a specific description of where the particles are. Counting up the number of atoms on each side gives us the macrostate. The macrostate describes the overall distribution of the atoms, for example, four atoms on the left and zero on the right (4:0), or three atoms on the left and one on the right (3:1). Table 9-1 shows all the possible microstates and macrostates for this experiment. With four particles, there are 16 possible microstates divided into five possible macrostates.

Table 9-1. All Possible Microstates and Macrostates of Four Atoms of Gas in a Two-Bulb Apparatus
Arrangement (Microstate)Distribution (Macrostate)
RYBGNone4:0
RYBG3:1
RYGB
RBGY
YBGR
RYBG2:2
BGRY
RB YG
YGRB
RGYB
YBRG
GRYB2:2
BRYG
YRBG
RYBG
NoneRYBG0:4

The most common macrostate is the 2:2 distribution; because of the 16 possible microstates, six will create this macrostate. Four microstates make up the 3:1 and 1:3 macrostates, making those less likely to occur. Much less likely to occur are the 4:0 or 0:4 macrostates, because only one in 16 microstates create them.

Even though the atoms are moving randomly, and even though no microstate is more likely than any other, the results show that the most likely macrostate is an equal distribution with two atoms on each side. In the language of physics, the more microstates that make up a macrostate, the more entropy the macrostate has. The more entropy a macrostate has, the more likely that macrostate is to occur. In equations, entropy is represented by the symbol S. Its units are Joules per mole (J/mol). In more general terms though, entropy is really a measure of the system disorder; the more particles we have, the more possible arrangements we have, and essentially the more mess we can make. Imagine a bedroom: Take out all the clothes and put them on the floor. The more clothes we own, the more of a disordered mess the floor of the room will be. The same is true of molecules.

The distribution of gas particles is just one way to see the effects of entropy. As we will see in the next section, entropy determines not only how particles arrange themselves in space, but also how heat flows between objects.

Glossary

Entropy

The amount of disorder in a system. Statistically, the more microstates that make up a macrostate, the more entropy that macrostate has.

Macrostate

In statistical mechanics, the overall distribution and thermodynamic properties of particles within a given system.

Microstate

In statistical mechanics, the specific distribution and thermodynamic properties of particles within a given system.

Statistical mechanics

Equations describing how particles distribute themselves in predictable ways, despite each particle moving around randomly.

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