2. A type of problem which is slightly more complicated in appearance, but exactly comparable in principle, is the following: "What is the factor for the conversion of a given weight of ferrous sulphate (FeSO_{4}), used as a reducing agent against potassium permanganate, into the equivalent weight of sodium oxalate (Na_{2}C_{2}O_{4})?" To determine the chemical equivalents in such an instance it is necessary to inspect the chemical reactions involved. These are: 10FeSO_{4} + 2KMnO_{4} + 8H_{2}SO_{4} --> 5Fe_{2}(SO_{4})_{3} + K_{2}SO_{4} + 2MnSO_{4} + 8H_{2}O, 5Na_{2}C_{2}O_{4} + 2KMnO_{4} + 8H_{2}SO_{4} --> 5Na_{2}SO_{4} + 10CO_{2} + K_{2}SO_{4} + 2MnSO_{4} + 8H_{2}O. It is evident that 10FeSO_{4} in the one case, and 5Na_{2}C_{2}O_{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}