I. Short Historical Introduction

In the late 1880's, Svante Arrhenius proposed that acids were substances that delivered hydrogen ion to the solution. He has also pointed out that the law of mass action could be applied to ionic reactions, such as an acid dissociating into hydrogen ion and a negatively charged anion.

This idea was followed up by Wilhelm Ostwald, who calculated the dissociation constants (the modern symbol is K_a. They are discussed elsewhere.) of many weak acids. Ostwald also showed that the size of the constant is measure of an acid's strength.

By 1894, the dissociation constant of water (today called K_w) was measured to the modern value of 1 x 10¯¹⁴.

In 1904, H. Friedenthal recommended that the hydrogen ion concentration be used to characterize solutions. He also pointed out that alkaline (modern word = basic) solutions could also be characterized this way since the hydroxyl concentration was always 1 x 10¯¹⁴ ÷ the hydrogen ion concentration. Many consider this to be the real introduction of the pH scale.

II. The Introduction of pH

Sörenson defined pH as the negative logarithm of the hydrogen ion concentration.

pH = - log [H⁺]

Remember that sometimes H₃O⁺ is written, so

pH = - log [H₃O⁺]

means the same thing.

So let's try a simple problem: The [H⁺] in a solution is measured to be 0.010 M. What is the pH?

The solution is pretty straightforward. Plug the [H⁺] into the pH definition:

pH = - log 0.010

An alternate way to write this is:

pH = - log 10¯²

Since the log of 10¯² is -2, we have:

pH = - (- 2)

Which, of course, is 2.

Another sample problem: Calculate the pH of a solution in which the [H₃O⁺] is 1.20 x 10¯³ M.

For the solution, we have:

pH = - log 1.20 x 10¯³

So you enter 1.20 x 10¯³ into the calculator, press the "log" button (NOT "ln") and then the sign change button (usually labeled with a "+/-").

The answer, to the proper number of significant digits is: 2.921. (I hope you took a look at the significant figures and pH discussion. If not, why don't you go ahead and do that right now. I can wait.)

Convert each hydrogen ion concentration into a pH. Identify each as an acidic pH or a basic pH.

1) 0.0015

2) 5.0 x 10¯⁹

3) 1.0

4) 3.27 x 10¯⁴

5) 1.00 x 10¯¹²

6) 0.00010

Sörenson also just mentions the reverse direction. That is, suppose you know the pH and you want to get to the hydrogen ion concentration ([H⁺])?

Here is the equation for that:

[H⁺] = 10¯^pH

That's right, ten to the minus pH gets you back to the [H⁺] (called the hydrogen ion concentration).

This is actually pretty easy to do with the calculator. Here's the sample problem: calculate the [H⁺] from a pH of 2.45.

The calculator technique depends on which type of button you have. Let's assume you have the standard key. It's labed EITHER x^y or y^x.

Some people have a calculator with a key labeled "10^x." In that case, enter the 2.45, make it negative, then press the "10^x" key. An answer appears!! Just remember to round it to the proper number of significant figures and you're on your way.