_{4}
in the other, each react with 2KMnO_{4}. These molecular
quantities are therefore equivalent, and the factor becomes
(10FeSO_{4}/5Na_{2}C_{2}O_{4}) or (2FeSO_{4}/Na_{2}C_{2}O_{4}) or
(303.8/134).

Again, let it be assumed that it is desired to determine the
factor required for the conversion of a given weight of potassium
permanganate (KMnO_{4}) into an equivalent weight of potassium
bichromate (K_{2}Cr_{2}O_{7}), each acting as an oxidizing agent
against ferrous sulphate. The reactions involved are:

10FeSO_{4} + 2KMnO_{4} + 8H_{2}SO_{4} --> 5Fe_{2}(SO_{4})_{3} +
K_{2}SO_{4} + 2MnSO_{4} + 8H_{2}O,

6FeSO_{4} + K_{2}Cr_{2}O_{7} + 7H_{2}SO_{4} --> 3Fe_{2}(SO_{3})_{3} +
K_{2}SO_{4} + Cr_{2}(SO_{4})_{3} + 7H_{2}O.

An inspection of these equations shows that 2KMO_{4} react with
10FeSO_{4}, while K_{2}Cr_{2}O_{7} reacts with 6FeSO_{4}. These are
not equivalent, but if the first equation is multiplied by 3 and the
second by 5 the number of molecules of FeSO_{4} is then the same in
both, and the number of molecules of KMnO_{4} and K_{2}Cr_{2}O_{7}
reacting with these 30 molecules become 6 and 5 respectively. These
are obviously chemically equivalent and the desired factor is
expressed by the fraction (6KMnO_{4}/5K_{2}Cr_{2}O_{7}) or
(948.0/1471.0).

3. It is sometimes necessary to calculate the value of solutions
according to the principles just explained, when several successive
reactions are involved. Such problems may be solved by a series of
proportions, but it is usually possible to eliminate the common
factors and solve but a single one. For example, the amount of MnO_{2}
in a sample of the mineral pyrolusite may be determined by dissolving
the mineral in hydrochloric acid, absorbing the evolved chlorine in a
solution of potassium iodide, and measuring the liberated iodine
by titration with a standard solution of sodium thiosulphate. The
reactions involved are:

MnO_{2} + 4HCl --> MnCl_{2} + 2H_{2}O + Cl_{2}
Cl_{2} + 2KI --> I_{2} + 2KCl
I_{2} + 2Na_{2}S_{2}O_{3} --> 2NaI + Na_{2}S_{4}O_{6}

Assuming that the weight of thiosulphate corresponding to the
volume of sodium thiosulphate solution used is known, what is the
corresponding weight of manganese dioxide? From the reactions given
above, the following proportions may be stated:

2Na_{2}S_{2}O_{3}:I_{2} = 316.4:253.9,

I_{2}:Cl_{2} = 253.9:71,

Cl_{2}:MnO_{2} = 71:86.9.

After canceling the common factors, there remains
2Na_{2}S_{2}O_{3}:MnO_{2}