# Acids, Bases, pH, pOH and the Math Behind Them

This post covers the math and mystery behind pH, pOH and the equilibrium constant for water.

Keynote presentation including log paper worksheet:  pH pOH and log paper activity

Learning Objective: to understand how pH and pOH add together. To understand that adding logs is equivalent to multiplying numbers.
Standard: Ca CH 5: Acids and Bases
Acids, bases, and salts are three classes of compounds that form ions in water solutions. As a basis for understanding this concept:
a. Students know the observable properties of acids, bases, and salt solutions.
b. Students know acids are hydrogen-ion-donating and bases are hydrogen-ion-accepting substances.
c. Students know strong acids and bases fully dissociate and weak acids and bases partially dissociate.
d. Students know how to use the pH scale to characterize acid and base solutions.
e. * Students know the Arrhenius, Br¿nsted-Lowry, and Lewis acid-base definitions.
f. * Students know how to calculate pH from the hydrogen-ion concentration.
g. * Students know buffers stabilize pH in acid-base reactions.

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Warm Up: {from Logs Lesson 1}  Describe how logarithms can be used to MULTIPLY two numbers.  You may write this in your own words or draw a sketch.  Hand in your index card when done.

The pH and pOH scales are a way of characterizing acids, bases and neutral water based solutions.
The pH and pOH scales depend on the bizarre fact that neutral water “self ionizes” or “dissociates” creating a small amount of H3O+  {Hydronium} and OH- {Hydroxide} ionic molecules.
In the1880’s, great scientists, including the brilliant Svante Arrhenius himself, thought that one water molecule broke down into H+ + OH-. But today we know the fact that 2 H2O molecules combine and “dissociate” into one  H3O+  {Hydronium} and one OH- {Hydroxide}.

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Concentrations and the self ionization constant for water

Concentrations are defined as the number of moles of a solute (the chemical being dissolved) dissolved in 1 liter of solvent (in this case water). This definition is known as “molar” concentration, abbreviated “M”. There is a similar definition, the number of moles of solute dissolved in 1 kg of solvent called “molal”, abbreviated “m” so don’t get confused by these two definitions.
Example: 3.545 grams of HCl in 1 liter of water yields a 0.1M solution.
(H{ 1 g / mol} + Cl {34.45 g / mol} )  / 10 = 0.1 mol HCl

The equilibrium constant is defined as the concentrations of a reaction’s products divided by the concentrations of the reaction’s reactants.
In this case, the equilibrium constant for water is given by;
Kw = [H3O+ ][OH-] / [H2O]2   = 1 x 10 -14
Since very little water is consumed in this dissociation, we can say that the concentration of water is essentially 1 and neglect the denominator.
We can also use the historical notation that Hydrogen is formed instead of Hydronium.
Kw = [H+ ][OH-]  =   1 x 10 -14
Since equal quantities of H+  and OH-  are produced, both concentrations must be equal.  [H+ ]  =   1 x 10 -7  and [OH-]  =   1 x 10 -7
Now by taking the negative of the logarithm of these numbers we get:
the “power” of H or pH  = 7  and
the “power” of OH or pOH  = 7

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pH and pOH Math Worksheet

Learning Objective: to understand how pH and pOH add together. To understand that adding logs is equivalent to multiplying numbers.
The worksheet has 6 strips of log paper. you will be using 2 strips for each pair.
Step 1: label the left side of 3 log paper strips starting at 100 upwards to 10-7. Label the remaining 3 strips starting at 10-7 to 10-14. You now have 3 pairs of log paper strips numbered from 100 to 10-14.
Step 2: Cut out the strips and tape each pair together at the 10-7 mark. You now have 3 long strips labeled from 100 to 10-14.
Step 3: Label the long strips “pH”,  ‘pOH” and “pH scale” respectively. Now label the right side of each strip, starting from the bottom “0” next to the 10-0 marking, “1” next to the 10-7 marking etc up to 14.
Step 4: use the  use the strips to add the log pH to the log pOH and to show that the product of pH and pOH equals 10-14.
Step 5: solve the problems on the next page.

To get the log paper strips, you can download and print this pdf or open the link to the keyote presentation at the top of this post.

7 decade log paper strips

### Warm Up: Day 2:{From Memory}

Write the balanced chemical equation for the self ionization of water.
Write the equation to calculate the equilibrium constant for water Kw.
Remeber that Kw = 1 x 10-14

PROBLEMS:

Show that for a neutral solution, pH = pOH   and [H+] * [OH-] = 10^-14  (Kw).
What values of pH, [H+], pOH, and [OH-] will satisfy this relation?
Show that for an acidic solution where pH = 5, pOH must     equal 9.
What are the concentrations of [H+] and [OH-] ?
Show that for a basic solution where pH = 11, pOH must equal 4.
What are the concentrations of [H+] and [OH-] ?
Why is the pOH number smaller for a basic solution, when the basic nature of the solution is due to an excess of OH- (hydroxide) molecules?

Now for the Goods!