**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.

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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