How Do Reactions Continue at Equilibrium and What Determines Their Completion?

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How do we know that the reaction continues during equilibrium? My textbook just gives the definition of equilibrium but offers no proof that the reaction has not stopped.

And why do some reactions go to completion but others don't? How can you tell if a rxn will go to completion or merely equilibrium?
 
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Reactions don't ever "stop" at equilibrium; products and reactants are just forming at a rate exactly equal to each other.

You could show that the reaction is still ongoing by LeChatelier's principle. If you remove one of the product, more will form.
 
One way to show that reactions keep going on at equilibrium is to add different isotopes that essentially behave identically to the other species, but can be distinguished by their mass. For example, if you consider an acid/base reaction:

A- + H+ <--> HA

You can add deuterated water (D2O) to the reaction without disrupting the equilibrium. After some time, you will find some amount of deuterated acid, showing that A is constantly exchanging hydrogens with the environment.

In certain systems, it is possible to follow the dynamics over time of a single molecule. In these systems, you can directly observe the fluctuations of a system at equilibrium. For example if a molecule can exist in one of two different states, you will see the single molecule stochastically transiting between the two different states.