Both molarity and normality are measures of concentration. One is a measure of the number of moles per liter of solution and the other changes depending on the solution's role in the reaction.
Molarity is the most commonly used measure of concentration. It is expressed as the number of moles of solute per liter of solution.
A 1 M solution of H2SO4 contains 1 mole of H2SO4 per liter of solution.
H2SO4 dissociates into H+ and SO4- ions in water. For every mole of H2SO4 that dissociates in solution, 2 moles of H+ and 1 mole of SO4- ions are formed. This is where normality is generally used.
Normality is a measure of concentration that is equal to the gram equivalent weight per liter of solution. Gram equivalent weight is a measure of the reactive capacitity of a molecule.
The solution's role in the reaction determines the solution's normality.
For acid reactions, a 1 M H2SO4 solution will have a normality (N) of 2 N because 2 moles of H+ ions are present per liter of solution.
For sulfide precipitation reactions, where the SO4- ion is the important part, the same 1 M H2SO4 solution will have a normality of 1 N.
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