Chemistry 30

2.4 The pH Scale

It is finally time to turn our attention to pH. pH is just another way to express [H⁺], the hydrogen ion concentration of an acidic or basic solution.

Hydrogen acid concentrations are often small numbers, such as 1.3 × 10^-3. pH is a method of transforming this number into something that is a little easier to work with.

In math class you may have learned about logarithms - log for short. We'll leave the definitions of logs to math and just work with how to find them here.

Get your calculators out. Different calculators work in slightly different ways, and it will be VERY IMPORTANT for you to know how to use yours when working with logs.

What about pH?

pH is defined as the negative log of hydrogen ion concentration. Mathematically it looks like this:	pH = - log [H+]
Because hydrogen ion concentrations are generally less than one (for example 1.3×10-3), the log of the number will be a negative number. To make pH even easier to work with, pH is defined as the negative log of [H+], which will give a positive value for pH.
Try the examples shown on the right. Find the pH, given [H+]. Answers are shown, but be sure you are able to arrive at that answer with your calculator! Notice the last example. 1.0 ×10-7 is the [H+] in pure water. Pure water therefore has a pH of 7.	[H+] pH 1. 1 ×10-3 3.0 2. 2.5 ×10-11 10.6 3. 4.7 ×10-9 8.3 4. 5.8 ×10-4 3.2 5. 1.0×10-7 7.0
By looking at the [H+] values in the table above, can you determine which solutions would be acidic, and which would be bases? Number 1 and 4 are acids. In those, [H+] is greater than 1.0×10-7 Numbers 2 and 3 are bases. In those solutions, [H+] is less than 1.0×10-7.
Working with negative powers of 10 is not easy for many of us, so some of you may be confused by trying to identify acids and bases based on [H+]. But if we use pH values instead we find it much easier to identify acids and bases. Acids pH < 7 The lower the pH, the stronger the acid Bases pH > 7 The higher the pH, the stronger the base Neutral solutions pH = 7 You can see some examples of the pH of various common substances here.

Examples of Calculating pH

1.	Calculate the pH of a 0.01M HNO3 solution?
	Solution: Begin finding pH by first finding [H+]. You should recognize HNO3 as a strong acid (check your Acid Table), meaning it breaks down completely into ions. Based on the balanced equation shown below we see that there is a 1:1 ratio between HNO3 and H+, so [HNO3] = [H+]: HNO3 (aq) → H+(aq) + NO3-(aq) [H+] = 0.01 pH = -log [H+]     = -log (0.01)     = - (-2.0)     = 2.0 answer
2.	Find the pH of a 0.01 M solution of ammonia. Ammonia is a weak base with Kb= 1.8 × 10-5
	Solution: This is a more difficult question because we have a weak base (it would be similar for a weak acid). Because weak acids and bases do not ionize completely, we must use Kb to determine ion concentrations. That will require a balanced equation. When writing equations for weak bases (that don't contain an OH–), be sure to include water, H2O as a reactant. Remember that the base will gain a hydrogen ion: NH3 (g) + H2O(l) ↔ NH4+(aq) + OH-(aq) Since ammonia is a base, first calculate [OH-].  Then use Kw to determine [H+] (see the previous section) Set up the Kb equation: Kb = [NH4+] [OH- ] [NH3]     Substitute values into the equation. Let x equal the unknowns 1.8 ×10-5 = (x) (x) 0.01     Rearrange the equation x2 = (1.8 × 10-5) (0.01)   x2 = 1.8 × 10-7 Take the square root x = 4.2 × 10-4   [OH-] = 4.2 × 10-4 M Next we calculate [H+]:   Kw = [H+] [OH-] Rearrange the equation [H+ ] = Kw [OH-] Substitute in known values and calculate [H+] [H+] = 1.0 ×10-14 4.2 × 10-4 [H+] = 2.4 × 10-11 Finally, convert [H+] into pH: [H+] = 2.4 × 10-11 pH = -log[H+]     = -log(2.4 × 10-11)   pH = 10.6 answer
pOH
	There is a way to simplify the last parts of this operation. In addition to pH, we can also define pOH: pOH = - log [OH-] For bases, once we find [OH-] for a base, we can quickly determine pOH: [OH-] =4.2 × 10-4 pOH = -log [OH-]     = -log (4.2×10-4)   pOH = 3.4 answer Next we make use of the following easy-to-memorize relationship: pH + pOH = 14 Does the number 14 ring a bell? Remember Kw = 1.0 × 10-14. The negative log of 1.0 × 10-14 = 14 Once we find pOH, it is a simple matter to find pH: pH + pOH = 14 OR pH = 14 - pOH       = 14 - 3.4     pH = 10.6 It doesn't matter which method you use to find [H+] and pH for a base - both will give you the same answer. Choose whichever method works best for you.

Finding [H⁺] when you know pH

Sometimes you need to work "backwards" - you know the pH of a solution and need to find [H⁺], or even the concentration of the acid solution. How do you do that?

To convert pH into [H+] involves taking the antilog of the negative value of pH .

[H+] = antilog (-pH)

As mentioned above, different calculators work slightly differently - make sure you can do the following calculations using your calculator. Practice as we go along . . .

Example. We have a solution with a pH = 8.3. What is [H⁺]?

With some calculators you will do things in the following order:

1. Enter 8.3 as a negative number (use the key with both the +/- signs, not the subtraction key)
2. Use your calculator's 2nd or Shift or INV (inverse) key to type in the symbol found above the LOG key. The shifted function should be 10^x.
3. You should get the answer 5.0 × 10^-9

Other calculators require you to enter keys in the order they appear in the equation.

1. Use the Shift or second function to key in the 10^x function.
2. Use the +/- key to type in a negative number, then type in 8.3
3. You should get the answer 5.0 × 10^-9

If neither of these methods work, try rearranging the order in which you type in the keys. Don't give up - you must master your calculator!

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Here are two final examples. Think carefully and determine what, exactly, you are asked to find and the steps needed to get there.

1.	Find the hydronium ion concentration in a solution with a pH of 12.6. Is this solution an acid or a base? How do you know?
	Solution: We can easily answer the second part of the question - the solution is a base because pH > 7. Next we need to find hydronium ion concentration, [H3O+], which you should remember will be the same as [H+] (section 1-3). To convert pH into [H3O+]: [H3O+] = antilog (-pH)   = antilog (-12.6)   [H3O+] = 2.5 × 10-13 answer
2.	A 0.24M solution of the weak acid, H2CO3, has a pH of 3.49. Determine Ka for H2CO3 (carbonic acid).
	Solution: This is much more difficult. In order to determine Ka  we need to know the concentrations of several things: [H+], [HCO3-], and [H2CO3] Begin with a balanced equation for the acid: H2CO3 (aq) ↔ H+(aq) + HCO3-(aq) Next set up the equilibrium expression, which will be needed to find Ka: Ka = [H+][HCO3-] [H2CO3]   Next find [H+] from pH. [H+] = antilog (-pH)   = antilog (-3.49)   [H+] = 3.2 × 10-4 The balanced equation tells us that [H+] = [HCO3-]. Substitute values into our Ka expression and solve: Ka = [H+][HCO3-] [H2CO3] The question gave us the the concentration of the acid, H2CO3, as 0.24 M   = [3.2 × 10-4][3.2 × 10-4] [0.24]       Ka = 4.3 ×10-7 answer

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These last two sections often prove to be the most difficult sections in the entire Chemistry 30 course for students. Be sure to follow the examples that have been provided carefully and make sure you understand every step. Doing the practice questions is a key to your success. Finish Practice Set 2 (questions 8 - 12), then complete Assignment 2.