Lesson 5: pH and Kw
When working with acidic or basic solutions, another way of expressing concentration is often used and that is pH. There is more to understanding pH than knowing its definition, so let's set the stage for pH by first looking at the self-ionization of water. After a definition of pH, we will then look at how the hydronium/hydroxide ion balance determines the acidity or basicity of a solution. The concentrations of hydronium and hydroxide ions in an aqueous solution are related to one another by the ionization constant for water, Kw. That constant can be used to calculate hydronium and hydroxide ion concentrations, and these in turn can be used to calculate pH and pOH values. Each of these topics can be found in the pages of this section.
Auto-Ionization of Water | pH Definition | Hydronium/Hydroxide Balance | Kw
Calculating Concentrations | Calculating pH and pOH
Auto-Ionization of Water
Water molecules collide with one another to cause the self-ionization reaction represented by this equation: |
2H2O  H3O+ + OH-
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It is a reversible reaction so the equation is usually written with the arrows going in both directions: |
2H2O  H3O+ + OH-
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pH Definition
pH can be viewed as an abbreviation for power of hydrogen or more completely, power of the concentration of hydrogen ion.
The mathematical definition of pH is a bit less intuitive, but with a calculator in hand, more useful. It says that the pH is equal to the negative log of the hydrogen ion concentration, or pH = -log [H+]. Using the Brønsted-Lowry approach that would be pH = -log [H3O+]. We'll come back to this definition when we start calculating pH values.
Hydronium/Hydroxide Balance
When an acid dissolves in water, additional H3O+ is formed, increasing the concentration of H3O+. For example, the concentration of H3O+ might be increased from 10-7 M up to 10-5 M. That is 100 times more concentrated. Note that the pH, the number behind the negative sign in the exponent, changes from 7 to 5. This is why acidic solutions have pH values lower than 7.
The acidity or basicity of a solution is related to the relative concentrations of H3O+ and OH-. If the concentration of H3O+ is more than the concentration of OH-, the solution is acidic. If the concentration of OH- is more than the concentration of H3O+, then the solution is basic. If the concentrations of H3O+ and OH- are equal to one another, the solution is neutral. |
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There is also an internal relationship between the concentrations of H3O+ and OH-. They are not independent of one another. As one goes up, the other
goes down. They cannot both go up because the higher concentrations of H3O+ and OH- would react with one another to make water molecules. That is a
consequence of the reversibility of the self-ionization reaction of water. (2H2O H3O+ + OH-)
Let's use the self-ionization of pure water as our starting point. The concentrations of both H3O+ and OH- are 1.0 x 10-7 M. (For your notes, write these and the following values down in the table in exercise 24.) If the concentration of H3O+ is doubled, the concentration of OH- willl be halved. If the concentration of H3O+ is halved, the concentration of OH- willl be doubled.
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If the concentration of H3O+ goes up by a factor of 10, to become 1.0 x 10-6 M, then the concentration of the OH- goes down by a factor of 10 become 1.0 x 10-8 M. If the concentration of H3O+ goes up by another factor of 10 to become 1.0 x 10-5 M, then the concentration of OH- goes down by another factor of 10 to become 1.0 x 10-9 M.
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This same pattern holds if the concentration of OH- is increased. Let's start again with neutral water. The concentrations of H3O+ and OH- are both 1.0 x 10-7 M. If the concentration of OH- is increased by a factor of 10 to become 1.0 x 10-6 M, then the concentration of H3O+ goes down by a factor of 10 to become 1.0 x 10-8 M. If the concentration of OH- is increased to 1.0 x 10-5 M, the concentration of H3O+ decreases to 1.0 x 10-9 M. |
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Kw
The relationship between the concentration of H3O+ and the concentration of OH- shows a very consistent pattern.
That pattern can be expressed mathematically by saying that the product of the concentration of H3O+ times the concentration of OH- remains a constant value of 1.0 x 10-14. That value has both a name and a symbol. It is variously called the ion product constant for water, the ionization constant of water,or simply the water constant. The symbol is Kw. You should write down this relationship in your workbook. |
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This equation for the ionization constant of water can be used in anumber of calculations, as you will see next.
Calculating Concentrations
The equation for the ionization constant of water can be used to calculate either the hydronium or hydroxide ion concentration if the other one is known.
As an example, let's calculate [OH-] given that [H3O+] = 4.5 x 10-5 M. Starting with the water constant equation [H3O+]·[OH-] = Kw, we can figure that [OH-] = Kw/[H3O+]. Then, substitute the known values for Kw and [H3O+] to get that [OH-] is equal to 1.0 x 10-14 divided by 4.5 x 10-5. That comes out to be 2.2 x 10-10 M for the concentration of hydroxide ion. |
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Practice
Now you figure out the missing values in this table (also shown in exercise 24-c in your workbook). Check your answers below and then continue. |
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Answers
The missing values are [H3O+] = 3.0 x 10-13 M and [OH-] = 7.1 x 10-5 M. |
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Calculating pH and pOH
Next let's look at values for pH and pOH. pOH is simply the power of hydroxide ion concentration and is figured the same way as pH but using the concentration of hydroxide ion instead.) For your notes, expand the table in exercise 24-a by making columns for pH and pOH.)
Let's start by working with the concentrations that are 1.0 x 10(raised to some power). These pH and pOH values can be figured very simply. When [H3O+] is 10-7 M, the pH is 7. Also the [OH-] is 10-7 M and the pOH is 7. Note that the pH and pOH add up to 14. |
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Now look at the acidic solutions. When the [H3O+] is 10-6 M, the pH is 6. Also, the [OH-] is 10-8 M and the pOH is 8. Again, the pH and the pOH add up to 14. When the [H3O+] is 10-5 M, pH is 5, [OH-] is 10-9 M, and pOH is 9. pH + pOH = 14. |
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Next, the basic solutions. When [OH-] is 10-6 M, the pOH is 6. Since [H3O+] 10-8 M, the pH is 8. When [OH-] is 10-5 M, and [H3O+] is 10-9 M, the pOH is 5 and pH is 9. Again in both cases the sum of pH and pOH is 14. |
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However, the hydrogen ion concentration is not always going to be equal to exactly 1 x 10 raised to a negative number. For example, we skipped over the value of 2.0 x 10-7. This is more complicated. However, if you use a calculator that will handle logarithms, it is a very simple calculation. First you enter the hydronium ion concentration. You can use decimal format or scientific notation. Next push the log button. Then change the sign by pushing the +/- button. In this case we get 6.70 for the pH. The other values can be obtained in the same way. (Your calculator may require you to follow a different sequence; if you need help, ask your instructor or get help in the lab.) |
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Significant Digits in pH
Now a word about significant digits and log values like pH's. Let's compare the parts of the pH value to the parts of the value in the scientific notation from which it was derived. The value of 2.0 x 10-7 shows both precision (two significant digits in 2.0) and size or magnitude (shown in the exponent -7). In the pH value, the size or magnitude is shown in the number in front of the decimal point. The two significant digits are shown after the decimal point. The size match is a little easier to see if the the concentration is written in nonstandard scientific notation. |
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Practice - pH
I'd like you to practice calculating pH values by figuring values for the two empty columns in this table (exercise 26 in your workbook). Write the hydronium ion concentrations in scientific notation and also calculate the pH value for each. Check your answers below, then continue with the lesson.
[H3O+] |
scientific notation |
pH |
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Answers
Your answers should be as follows:
| [H3O+] | scientific notation |
pH |
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Note that each pH value has one digit to the right of the decimal point because each concentration value had one significant digit.
If your answers are not correct, get some help from the instructor.
Hydronium Ion Concentration from pH
It is also possible to run the calculations in the other direction as well. If you know the pH, you can calculate the hydrogen ion concentration. Here is how you do it. Let's say the pH is 4.3. Enter the pH value and make it negative using the +/- button. Then press the 10x button. It will probably be the 2nd function or inverse of the log button. I get 0.00005 or 5 x 10-5 M. If you don't get this value when you try it with your calculator, or if you cannot find the right buttons, check with the instructor in the lab.
Practice - pH and pOH
Once you have that calculation under control, try your hand at filling in the blank spots in this table (exercise 27 in your workbook). Check your answers below then continue with the lesson.
| pH | [H3O+] | pOH | [OH-] |
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2.3 x 10-5 |
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7.8 x 10-4 |
Answers
All concentrations are expressed in molarity.
| pH | [H3O+] |
pOH | [OH-] |
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0.0006 or |
10.8 |
1.6 x 10-11 or |
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0.000000006 or |
5.8 |
1.6 x 10-6 or |
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2.5 x 10-8 |
6.40 |
4.0 x 10-7 |
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0.0050 |
11.70 |
2.0 x 10-12 |
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2.0 x 10-12 |
2.30 |
0.0050 |
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4.3 x 10-10 (from the Kw equation) or 4.4 x 10-10 (if going from pH to conc.) |
4.64 |
2.3 x 10-5 |
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7.8 x 10-4 |
10.89 |
1.3 x 10-11 |
If you had trouble with any of these, first double check your calculations to make sure you entered the values and functions correctly on your calculator. Then check with your instructor.