100 mL of a given hard water (density 1 g/mL) needs $31.2 \mathrm{mL}\text{ of }0.04\mathrm{N} {\mathrm{H}}_2{\mathrm{SO}}_4$ for complete reaction The hardness of water will be

Milliequivalents of H₂SO₄ are calculated as:

$\begin{array}{ccc}\mathrm{Milliequivalents} \mathrm{of} {\mathrm{H}}_2{\mathrm{SO}}_4& =& \mathrm{Volume} (\mathrm{in} \mathrm{mL})\times \mathrm{Normality}\\ & =& 31.2 \mathrm{mL}\times 0.04 \mathrm{N}\\ & =& 1.248\end{array}$

Milliequivalents of H₂SO₄ are equal to the hardness of the water. Hence, milliequivalents of CaCO₃ are equal to the milliequivalents of H₂SO₄. 

The expression that shows the relation between milliequivalent and equivalent weight is:

$\mathrm{Milliequivalent}=\frac{\mathrm{weight}}{\mathrm{equivalent} \mathrm{weight}}$

Use the above expression to determine the weight of CaCO₃.

$\begin{array}{ccc}\mathrm{weight} \mathrm{of} {\mathrm{CaCO}}_3& =& \mathrm{milliequivalent}\times \mathrm{equivalent} \mathrm{weight}\\ & =& \mathrm{milliequivalent}\times \frac{\mathrm{molecular} \mathrm{weight}}{\mathrm{n}-\mathrm{factor}}\\ & =& \frac{1.248}{1000}\times \frac{100}{2}\\ & =& 0.0624 \mathrm{g}\end{array}$

The hardness of water is measured at 10⁶ (or ppm).

 $\begin{array}{ccc}\mathrm{Hardness} \mathrm{of} \mathrm{water} (\mathrm{or} \mathrm{equivalent} \mathrm{of} {\mathrm{CaCO}}_3)& =& \frac{\mathrm{Weight} \mathrm{of} {\mathrm{CaCO}}_3}{\mathrm{Desnity} \mathrm{of} {\mathrm{CaCO}}_3\times \mathrm{Volume} \mathrm{of} {\mathrm{CaCO}}_3}\times 106\\ & =& \frac{0.0624 \mathrm{g}}{100\times 1}\times 106\\ & =& 624 \mathrm{ppm}\end{array}$