Deriving the Henderson-Hasselbach Equation

Many students are familiar with the ICE (Initial, Change, Equilbrium) Table when calculating the pH of a weak acid (or weak base) solution. The Henderson-Hasselbach equation will also enable you to calculate the pH of a weak acid (or weak base solution). In this post, I will derive the Henderson-Hasselbach equation from the equilibrium expression. In my next post, I will discuss the similarities and differences between using an ICE table versus using the Henderson-Hasselbach equation.

The first step in deriving the Henderson-Hasselbach equation is to set up your equilibrium expression. In this example, we will look at the equilibrium of the weak acid, hydrofluoric acid.

Our next steps are mathematical in nature.  Multiply both sides by [HF] and then divide both sides by [F-] to get H_3o^+ by itself.

The next step is to take the negative logarithm of both sides.

For the remainder of the post, I am just going to work with the yellow highlighted section of the above equation.  We are going to simplify the equation and get to a pK_a value.  By our laws of logarithms, when two values are multiplied, it is the same as distributing the log to both values and taking the sum.  See below.

A little more manipulation and we can rearrange the part of the equation highlighted in blue.  Here you will see me using the exponent rule x^{-n}=\frac{1}{x^n}

Now a logarithm rule:  log_a x^b = b log_a x

After these manipulations we can return to our yellow highlighted equation.  Our final step is to simplify our expression so that we are using pH and pK_a.

And voila!  The Henderson-Hasselbach equation derived from the equilibrium expression of hydrofluoric acid.



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