pH Scale

Logarithms and pH Calculations

Introduction to pH and Logarithms

In this section of the logarithms tutorial you will learn to use logarithms to solve common math problems involving pH. To introduce this topic you will first learn about what pH is, what it means in Chemistry, why logarithms are used to quantify pH, and why pH is of interest to chemists, scientists, and mathematicians.

What is pH?

pH is the negative base-ten logarithm of the hydronium ion concentration in a fluid, measured in moles per liter:

pH = -log[H3O+]

The molar concentration of hydronium in a fluid is typically represented with the notation [H3O+].

Hydronium ion concentrations in a fluid can range between 0.00000000000001 (10^-14) and 1.0 (10^0) moles per liter. These numbers are small and differ by a factor of 10^14. Logarithms and the pH scale are used to represent these numbers because the notation is more compact:

1.0 can be represented as 1.0 * 10^0, for a pH of 0.

0.00000000000001 can be represented as 1.0 * 10^-14, for a pH of 14.

The pH scale is a compact way to  represent these minute and widely ranging molar concentration values. An increase of 1 on the pH scale equates to a factor of ten decrease in [H3O+]. A decrease of 1 on the pH scale equates to a factor of 10 increase in [H3O+]

pH is value between 0 and 14, inclusive, indicating the acidity of a fluid:

Values less than 7 indicate a fluid is acidic and the hydronium ion concentration is less than 1.0 * 10^-7 moles per liter.

Values greater than 7 indicate a fluid is basic, and the hydronium ion concentration of more than 1.0 * 10^-7 moles per liter.

A pH of 7 means a solution is acid neutral and the hydronium ion concentration [H3O+] equals the hydroxide ion concentration [OH-] of 1.0 * 10^-7 moles per liter

All aqueous solutions contain hydronium (H3O+) and hydroxide (OH-), and the product of their respective molar concentrations is 1.0 * 10^-14 moles per liter.
H3O+ is an extra hydrogen ion bonded to a water molecule. OH- is a water molecule that lost a hydrogen ion.

pH is considered by chemists and scientists to be an important measure of water purity. Water is amphoteric. It can act as an acid and as a base, in a process called autoionization. Water molecules (H2O) thow off hydrogen ions, H+, and produce hydronium (H3O+) and hydroxide (OH-) ions. Greater concentrations of impurity in water triggers more autoionization. The more pure the water, the lower the ion concentration. Adding an acid to pure water disturbs the equilibrium and raises the H3O+ concentration. Adding a base raises the OH- concentration.

• In a neutral solution:  [H3O+]  =  [OH-] = 1.0 *10^-7 moles/liter, pH = 7
• In an acidic solution:  [H3O+]  >  [OH-] , pH < 7
• In a basic solution:     [H3O+]  <  [OH-], pH > 7

Two Key Equations:

There are two key equations to understand and use with logarithms and pH

pH = -log[H30+], meaning pH is the negative base ten logarithm of the hydronium ion concentration, and
[H3O+] = 10^(-pH), meaning hydronium ion concentration is ten raised to the power of negative of pH

pH math problems will typically present either pH and ask for computation of [H3O+], or will present a value for [H3O+] and ask for the pH. Knowing how to algebraically convert between the two key equations is essential in solving pH math problems. Conversion between these two equations is done as follows:

pH = -Log([H3O+])
-pH = Log([H3O+]) .................. flip the negative signs
10^-(pH) = 10^(Log[H3O+]) ............. fourth property of logarithms. See section 1 for review
10^-(pH) = [H3O+] .......................... Ten raised to the power of the base ten log of a number is that number
[H30+] = 10^-(pH)

Usage note: Most math and chemistry books offer a rule regarding significant figures and decimal positions. If, for example, there are three significant figures on the logarithm value, use three decimal positions on the pH.

Example 1:

If the hydronium ion concentration in water is 2.5 * 10^-6 moles per liter then what is the pH?

[H3O+] = 2.5 * 10^-6 M/L
pH = -Log([H3O+])
pH = -Log(2.5 * 10^-6)
pH = -[Log(2.5) + log(10^-6)]
pH = -[0.398 - 6] ~= + 5.6

This means that if [H3O+] = 2.5 * 10^-6 then pH is ~ +5.6

To verify this enter 10^-5.6 in a calculator and note the result is about 2.5 * 10^-6.

Example 2:

If the pH of diet soda is 3.12 then what is the hydronium concentration [H3O+]?

pH = -Log[H3O+]
3.12 = -Log[H3O+]
10^3.12 = 10^(-Log[H3O+])
10^-3.12 =[H3O+]                  Careful handling the negative sign
[H3O+] = 7.59 * 10^-4 moles/liter             Do this on a calculator

This means if the pH of a soda is 3.12 then the hydronium concentration is .000759 moles/liter, also represented as 7.59 * 10^-4 moles per liter.

To verify this as correct enter 10^-3.12 on a calculator and note the result is 7.59 * 10^-4

Example 3:

If the Hydronium ion concentration [H3O+] is 0.0032 moles per liter then what is the pH?

[H3O+] - 0.0032
pH = -log[H3O+]
pH = -[(Log (3.2 * 10^-3))]
pH = -[log(3.2) + log(10^-3)]
pH = -[0.505 - 3] = 2.49

This means that if [H3O+] is 3.2 * 10^-3 moles per liter then the pH is about 2.49.

To verify this enter 10^-2.49 in a calculator and note the result is about 3.2 * 10^-3.

Example 4:

If the pH of a fluid is 8.3 then what is the Hydronium ion concentration?

pH = -Log[H3O+]
pH = 8.3
8.3 = -Log[H3O+]
10^8.3 = -[10^(Log([H3O+]))]
10^-8.3 = [H3O+]
5.01 * 10^-9 = [H3O+]

This means that if pH is 8.3 then the  hydronium ion concentration [H3O+] is 5.01 * 10^-9 moles per liter.

To verify this enter 10^-8.3 in a calculator and note the result is about 5.01 * 10^-9.

Example 5:

If a fluid sample has a Hydronium ion concentration [H3O+] of 2.5 * 10^-6 moles per liter then what is the pH?

[H3O+] = 2.5 * 10^-6
[H3O+] = 10^-pH
Log([H3O+]) = Log(10^-pH)
Log([H3O+]) = -pH
pH = -Log ([H3O+)] = -log(2.5 * 10^-6) = -(Log(2.5) + Log (10^-6))
pH = -0.3979 + 6 = +5.6

This means that if the hydronium ion concentration is 2.5 * 10^-6 moles per liter then pH is about 5.6.

Verify this by entering 10^-5.6 in a calculator and note the result is about 2.5 * 10^-6.

Note on sources

Most of the technical information for this section is from:

Kotz, John C., Treichel, Paul M., and Townsend, John T. Chemistry & Chemical Reactivity. Thomson Brooks Cole, 2009.

Tro, Nivaldo J. Chemistry Structure and Properties. Pearson, 2015.

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