The Law of Mass Action

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To start, consider the following scenario: a tank of water is separated into two compartments that are connected by a small opening that allows water to move from one compartment to the other. With the opening blocked, an excess of water is added to one side, making the water levels in the tank different from each other, with one side having a higher water level than the other. What happens when the opening between the two sides is unblocked? Common sense tells us that the water will flow from the side with the higher level to the one with the lower level. Why? Such a flow of water decreases the potential energy of the entire system by decreasing the height of its center of mass.

Now, think about how fast the water will flow through the opening. Will it flow at a constant rate and then stop when the two levels are equal? Or will the flow be relatively fast at first and then taper off as the difference in water levels decreases? Your intuition probably is suggesting the latter, correctly. It just "feels" right that as the two levels get closer and closer together, the force driving the water from one side to the other should get smaller and smaller, meaning it can't "push" the water through the opening as strongly, and the rate of flow decreases accordingly. But what is the underlying mechanism for this decrease in the rate of flow? So when the two water levels eventually become the same, what happens then? Does the movement of water stop completely. By all appearances, it does, but we can test that hypothesis as explained below.

If we repeat the above sequence of events but add a different color of dye to each side, we get a better picture of what happens before and after the water reaches the same level in the two sides. In the figure, the water dyed blue (on the left) is initially at a higher level than that red (on the right). Opening the channel between the sides results in a plume of blue water entering the right side, displacing the red water initially. When the two sides become equal there are still regions of red dye and blue dye. If there was no movement of water (or dye molecules) from that point forward, the distribution of dye would remain, i.e., more blue on the left and more red on the right. This is not would happen, at least over an extended period of time. Because molecules are constantly in motion, the dye molecules will eventually become evenly dispersed on both sides, making the two compartments the same shade of purple. This is a result of diffusion, the random motion of molecules: over time the excess of blue dye on the left side and the excess of red dye on the right give way to an equal concentration of the dyes. When the two two shades of purple are the same, the following two conditions must be true:

\[ \text{[Blue]}_{left} = \text{[Blue]}_{right} \nonumber \]

\[ \text{[Red]}_{left} = \text{[Red]}_{right} \nonumber \]

Here the terms [Red] and [Blue] refere to the concentrations of the dyes. Since the dye concentrations are not the same on the two sides initially, even after the two water levels become equal, in order to achieve equal concentrations in the two compartments the dye molecules must be able to pass through the opening as described by the following equations:

\[ \ce{Blue_{left} <=> Blue_{right}} \nonumber \]

\[ \ce{Red_{left} <=> Red_{right}} \nonumber \]

Getting back to the above hypothesis, we now have a an answer to the experiment. Because water molecules should behave the same way as dye molecules, qualitatively at least, we would expect the following equation to also be operative in the system:

\[ \ce{H2O_{left} <=> H2O_{right}} \nonumber \]

The movement of water does not stop when the two sides of the compartment attain equivalent levels of water. Rather, the net flow of water from one side to the other approaches zero as the two levels approach one another, but that is different - in a profeound way - from they movement of water stopping.

Section Outline:

0. No dyes; water levels reaching equilibrium; is it static?

1. Physical Mixing of Dyes

2. Quantitative Expression

3. Law of Mass Action

4. Summary & Properties of Chemical Systems at Equilibrium (and a brief explanation of physical systems at equilibrium)