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PPM means per part million or milligram per litre of solution.Calculation of ppm for a given compound say $x \pu{gms}$ of $\ce{Ca(HCO_3)}$ in 1 liter of water would be done like

$$ppm= \frac{1000\cdot x}{1\pu{1L}}$$

But I do not understand by what's meant by in terms of $\ce{CaCO3}$ is it equivalents or weight? This was asked in an exam and the formula is not covered in my textbook.

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It is meant as molar equivalent. $\ce{CaCO3}$ has the molar mass about $\pu{100 g / mol}$, $\ce{Ca(HCO3)2}$ has the molar mass cca $\pu{ 162 g / mol}$. Therefore $\pu{100 g}$ of the former is equivalent to $\pu{ 162 g }$ of the latter. The similar for the concentration expressed in ppm. E.g. if water contains $\pu{1.62 ppm}$ of $\ce{Ca(HCO3)2}$, it is equivalent to $\pu{1 ppm}$ of $\ce{CaCO3}$

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Expressing the hardness of water by ppm CaCO3 is nothing but a conventional convenience in water industry. There is a background behind it.

Basically, you titrate hard water with ethylenediamminetetraacetate (EDTA) and the moles of EDTA consumed are assumed to be consumed by Ca2+ ions only. As you would guess, this is not true because hard water contains many other ions which can form a complex with EDTA especially magnesium is always present. Anions have nothing to do with the calculation of ppm CaCO3. The calculation is rather trivial

moles of EDTA used = moles of Ca2+ (as calcium forms 1:1 complex with EDTA) = moles of CaCO3 and even = moles of Ca(HCO3)2.

One can easily convert moles of CaCO3 into parts per million from the known volume of hard water taken from titration.

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