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Fifteen conversions between pH, pOH, [H_{3}O^{+}], and [OH¯]

**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 a 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**

You may benefit by reading the Sörenson article introducing pH. By the way, the p stands for the German word 'potenz,' which means strength.

Sörenson defined pH as the negative base-10 logarithm (shown this way: log_{10}) of the hydrogen ion concentration.

pH = −log [H^{+}]

Remember that sometimes H_{3}O^{+} is written, so

pH = −log [H_{3}O^{+}]

means the same thing.

**Example #1:** The [H^{+}] in a solution is measured to be 0.010 M. What is the pH?

**Solution:**

1) Plug the [H^{+}] into the pH definition:

pH = −log_{10}0.010

2) An alternate way to write this is:

pH = −log_{10}1.0 x 10¯^{2}

3) Since the log_{10} of 10¯^{2} is −2, we have:

pH = −(−2)pH = 2.00 (two sig figs)

Let's discuss significant figures and pH.

In the first example, I used log_{10}. I'm going to stop doing that and simply write log. However, a warning:

Some textbook authors (especially in math) use log to indicate the natural logarithm, NOT the base-10 logarithm. As you go on in chemistry, the use of the natural logarithm (symbolized this way: ln. That's the letter ell, not the numeral one) is used more often. Be aware that a chemistry author may choose to use "log" instead of "ln."

**Example #2:** Calculate the pH of a solution in which the [H_{3}O^{+}] is 1.20 x 10¯^{3} M.

pH = −log 1.20 x 10¯^{3}This problem can be done very easily using your calculator. However, be warned about putting numbers into the calculator.

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 "+/-").pH = 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.)

For the examples below, convert each hydrogen ion concentration into a pH. Identify each as an acidic pH or a basic pH.

**Example #3:** 0.0015 M

pH = −log 0.0015pH = −(−2.82)

pH = 2.82

acidic

**Example #4:** 5.0 x 10¯^{9} M

pH = −log 5.0 x 10¯^{9}pH = −(−8.30)

pH = 8.30

basic

**Example #5:** 1.0 M

pH = −log 1.0pH = −(0)

pH = 0.00

acidic

Yes, a pH of zero is possible, it is just uncommon. In fact, watch out for this teacher test trick. What's the pH when [H

^{+}] = 2.0 M? That's right, NEGATIVE 0.30. It is possible to have a negative pH, it is just uncommon to see them.

**Example #6:** 3.27 x 10¯^{4} M

pH = −log 3.27 x 10¯^{4}= 3.485acidic

**Example #7:** 1.00 x 10¯^{12} M

pH = −log 1.00 x 10¯^{12}= 12.000basic

**Example #8:** 0.00010 M

pH = −log 0.00010 = 4.00acidic

Sörenson also 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.

**Example #9:** Calculate the [H^{+}] given a pH of 2.45.

1) The calculator technique depends on which type of calculator button you have. The following instructions assume you have a key labeled EITHER x^{y} or y^{x}.

(a) Enter the number "10" into the calculator. (Do NOT then press the EXP or EE key.)

(b) Press the x^{y}(or the other, depending on what you have)

(c) Enter 2.45 and make it negative with the +/- key.

(d) Press the equals button and the calculator will do its thing.

2) The following instructions are for a calculator with a key labeled "10^{x}."

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.

3) One more comment about the way the answer appears on the calculator. The two most common ways for the answer to appear are:

3.548133892E-3 or 0.003548133892That E-3 means this:

x 10¯

^{3}<--- that x is a times signThe final answer (to the proper number of significant figures) is

[H

^{+}] = 3.5 x 10¯^{3}M or 0.0035 MNotice the inclusion of the M for molarity.

Go to a similar discussion about pOH

Fifteen conversions between pH, pOH, [H_{3}O^{+}], and [OH¯]