[sodium.dioxid]

HF + H2O F- + H3O+
F- + H2O FH + OH-
H2O + H2O   H3O+ + OH-

First, are all of these three reactions occurring simultaneously?
Second, isn't it too complicated to make sense and keep track of all of these at once? I am having a hard time understanding how exactly this three-way system achieves equilibrium. I can't seem to "imagine the process" of getting to equilibrium. I can't play it out in my head.

[Borek]

There are only two independent reactions in this system. Whether all three occur I don't know, but if any two are present, system will behave as if the third one was present as well.

You can easily check that equilibrium constant for the third reaction can be calculated using equilibrium constants for two other.

[juanrga]

HF + H2O F- + H3O+
F- + H2O FH + OH-
H2O + H2O   H3O+ + OH-

First, are all of these three reactions occurring simultaneously?

Yes and others such as

2HF H₂F⁺ + F^-
F^- + HF HF₂^-
HF₂^- + HF H₂F₃^-

Depending of the amounts of HF and H₂O some reactions will be more relevant than others.

Second, isn't it too complicated to make sense and keep track of all of these at once? I am having a hard time understanding how exactly this three-way system achieves equilibrium. I can't seem to "imagine the process" of getting to equilibrium. I can't play it out in my head.

Each reaction achieves equilibrium by itself and the global equilibrium state can be though of as a superposition of each individual equilibrium.

[sodium.dioxid]

There are only two independent reactions in this system.

Can you please elaborate a little more on this? This statement is interesting.

[Borek]

Let's assume there are only two reactions in the solution:

H₂O H⁺ + OH^- (K_w)

HF H⁺ + F^- (K_a)

When you add some fluoride F^- it will react with H⁺, removing it from the solution, thus it will shift equilibrium of the first reaction. As long as we are observing only the outcome of the reaction (concentrations of reagents), we can't differentiate between the two step reaction described above and reaction

F^- + H₂O HF + OH^- (K_b)

Write equilibrium constants for all three reactions and check that K_w=K_aK_b. There are only two independent reactions in the system - doesn't matter which two you will select, you can always write the third equation and calculate the third equilibrium constant.

[sodium.dioxid]

That was very insightful; thank you.