# The Relationship Between Dilution and Dissociation

December 7, 2016

## H: Some Definitions

Be advised the equilibrium constant, $K$The equilbrium constant is some number, the value of which doesn’t really matter for today’s purpose, but that represents the position of equilbrium for the reaction. , of a weak acid:

where $\mathrm H^+$ is the hydrogen ion, $\mathrm A^-$ is the conjugate base, and $\mathrm H\mathrm A$ is the weak acid. If $x$ is the concentration of the hydronium, It follows that the concentration of hydronium is equivalent to that of the conjugate base. where

then because $\mathrm H\mathrm A$ is a weak acid,

Where [HA]0 is the initial acid concentration

Exciting stuff, I know! But here we go: there are two important things to note here:

1. The equilbrium constant is roughly the square of the hydronium concentration, divided by initial acid concentration.
2. The reaction of the weak acid can be said to have some parameter ∂, where ∂ is a constant representing the percent of the acid dissociated at equilibrium.

Okay. We now have everything we need, so consider the following:

## He: What will happen to ∂ if we dilute the solution?

Hastily, most will say ,

Well, that’s obvious, clearly the percent dissociation must decrease!

Why?

If it’s diluted, the solution is less concentrated of hydronium ions, so evidently a lesser percentage of the acid dissociates.

With only a minute of thought this makes sense - a sort-of ‘duh’ derivation.

Unfortunately, it’s completely wrong.

## Li: Percent dissociation is inversely proportional to the concentration.

With a minute of thought, we can understand why this is - first intuitively, then mathematically. When we dilute an acid, we are essentially adding more area upon which the acid can react, thus leading to a higher rate of dissociation. To prove this mathematically:

Let’s say we dilute our solution by a factor of ten. The reaction quotient $Q$ (like $K$, but not at equilibrium) can then be described as

Because the reaction quotient is far to the left of the equilbrium constant, the reaction will shift to the right The acid will dissociate further, increasing the numerator of the reaction quotient. in order to reach equilibrium.

QED - diluting a weak acid shifts the dissociation reaction to the left, leading the acid to dissociate more in order to reestablish equilibrium, thus increasing the percent dissociation.

Phew. Easy!

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