## What Does pH Mean in a Buffer?

In chemistry, pH is a measure of the hydrogen-ion concentration in a solution . The pH of a buffer can be calculated from the concentrations of the various components of the reaction. As mentioned in the previous unit, the balanced equation for a buffer is:

The strength of a weak acid (buffer) is usually represented as an equilibrium constant.
The acid-dissociation equilibrium constant (K_{a}), which measures the propensity of an acid to dissociate, for the reaction is:

For more information on equilibrium constants or acid-dissociation constants, see the respective units.
The greater the value of K_{a}, the more the formation of H^{+} is favored, and the lower the pH of the solution.

## Calculating pH of a Buffer

The pH of a solution of a weak acid depends on the strength of the acid and the other components in the solution.
In the simplest case, the weak acid is the only compound in water.
In this case, the pH can be found using the concentration of the acid, K_{a}, to solve for concentration of H^{+}.

Applying the equilibrium to the acid-association expression yields:

Rearranging the above yields:

which can be solved for x using the quadratic equation.
At equilibrium, x is the concentration of [H^{+}].
Therefore, pH can be solved using the following equation:

## A Simplified Method for Calculating Buffer pH

In cases where [HA] is more than 1000 times greater than K_{a}, the acid will not deprotonate much, and the value of x will be small.
Therefore, [HA] - x ≈ [HA].
This simplifies the K_{a} expression to:

Solving for x yields:

Then the pH can be solved as above. The formulas can be combined to give:

but only if [HA] >> K_{a}.

## Example of Calculating the pH of Buffer Solution

Two buffer solutions are made from an acid HA with a K_{a} = 1 x 10^{-5}.
One solution has a concentration of 0.10 M, and the other has a concentration of 5 x 10^{-4 }M.
Calculate the pH for both solutions using both methods described above.

### SOLUTION:

*0.1 M Solution*

Using the full method gives us the following quadratic:

which yields x = 9.95 x 10^{−3} M and pH = 3.00.

The simplified method gives us the following:

In this example, [HA] is more than 1000 times greater than K_{a}, and both methods yield the same result.

*0.5 x 10 ^{−4 }M Solution*

Using the full method gives us the following quadratic:

which yields x = 6.6 x 10^{−5} M and pH = 4.18.
The simplified method gives:

Here, the results differ by 0.03 pH units.
As [HA] becomes closer in value to K_{a}, then the difference will increase even more.

## ICE Tables: A Useful Tool For Solving Equilibrium Problems

ICE (initial, change, equilibrium) tables are very helpful tools for understanding equilibrium, and as such, for calculating the pH of a buffer solution. Consider, for example, the following problem:

Calculate the pH of a buffer solution that initially consists of 0.0500 moles of NH_{3} and 0.0350 moles of NH_{4}^{+}, after 30.0 mL of 0.50 M NaOH has been added to the buffer.
Note: K_{a} for the NH_{4}^{+} is 5.6*10^{-10}.

We know that initially there are 0.0350 moles of NH_{4}^{+} and 0.0500 moles of NH_{3}.
We also know that OH^{-} will react with NH_{4}^{+} as:

The decrease in moles of NH_{4}^{+ }will be equal to the increase in NH_{3}, and both will equal the moles of NaOH added.
We can therefore determine all three values with one calculation, that of moles of NaOH:

Thus, the change in moles of NH_{4}^{+} is -0.015 mol and the change in moles of NH_{3} is +0.015 mol.
The equilibrium amounts of each can be calculated as the sum of initial and change, leaving 0.0650 moles of NH_{3} and 0.02 moles of NH_{4}^{+}.
Throughout this whole process we have ignored the change in concentration of H^{+}, which is what we need to find pH.
This change we will simply define as x, which along with all the initial, change, and final information we can fit into a table.

Sample table:

Depending on what is asked by a problem, x can be inserted into any cell in an ICE table, which helps organize data and makes it easier to visualize how to find the unknown variable. In this example, we can find x using the equilibrium concentrations: