What is the oxidation number of iron in goethite?

Goethite is a dark and usually uninteresting mineral.

Its chemical formula is #"FeO(OH)"#, although it often contains small amounts of other metals such as manganese.

We can use the rules for oxidation numbers to determine the oxidation number
of #"Fe"#.

The essential rules for this problem are:

1. The oxidation number of oxygen in a compound is almost always -2.
2. The sum of the oxidation numbers of all atoms (or ions) in a neutral compound
   is 0.
3. The sum of the oxidation numbers of all atoms in an ion equals charge on the ion.

Per Rule 1, the oxidation number of #"O"# is -2.

Write the oxidation number above the #"O"# in the formula:

#"Fe"stackrelcolor(blue)("-2")("O")"(OH)"#

Per Rule 3, the sum of all the oxidation numbers in the hydroxide ion equals -1.

Put -1 above the #"OH"#.

#"Fe"stackrelcolor(blue)("-2")("O")stackrelcolor(blue)("-1")("(OH)")#

Per Rule 2, the sum of all the oxidation numbers in #"FeO(OH)"# (the blue numbers below the formula) must equal zero.

Let #x =# the oxidation number of #"Fe"#. Then

#x-2-1 = 0#

#x = 2+1 = 3#

There is only one #"Fe"# atom, so its oxidation number must be +3.

Write the oxidation number above the #"Fe"#.

#stackrelcolor(blue)("+3")("Fe")stackrelcolor(blue)("-2")("O")stackrelcolor(blue)("-1")("(OH)")#

The oxidation number of #"Fe"# is +3.