Question

A 500 ml sample of the effluent from a water softener required 6 drops of standard soap solution to produce a permanent lather. The soap solution had been calibrated against an artificial hard water solution containing 0.136 g of $CaC{{l}_{2}}$ per litre. On the average it required 28 drops of standard soap solution to lather 500 ml of the artificial sample. Then the hardness of effluent sample in terms of ppm of $CaC{{O}_{3}}$ is
(A) 123 ppm
(B) 26 ppm
(C) 0.123 ppm
(D) 38 ppm

Answer 1

Simply we can see that here is comparison of a standard solution and a sample solution with respect to their respective hardness. So, first we should have a basic idea about the hardness of water then use the knowledge to calculate the same.

Complete answer:
Let us know about what we mean by hardness of water,
Hardness of water-
The amount of dissolved magnesium and calcium in the water defines the hardness of water. In short, the water highly dissolving minerals in it is definitely hard in nature.
Hardness of water is measured in grains per gallon, in mg of Ca per litre or ppm (parts per million). The main cause of water hardness is the presence of divalent ions in water; Ca++, Mg++, and Fe++ especially.
The more accurate result when we determine the hardness of water is given by EDTA titration. The hardness can be removed by several techniques such as distillation, partial freezing, ion exchange or reverse osmosis.
Now, let us see the illustration,
According to given data we can say that,
Hardness of effluent sample = $\dfrac{6}{28}$ hardness of $CaC{{l}_{2}}$solution
As we know,
Mass of $CaC{{l}_{2}}$ = 111 gm/mol
Mass of $CaC{{O}_{3}}$ = 100 gm/mol
i.e. 111 parts of $CaC{{l}_{2}}$ = 100 parts of $CaC{{O}_{3}}$
so, by the given data,
0.136 parts of $CaC{{l}_{2}}$ = $\dfrac{100}{111}\times 0.136$ = 0.123 parts of $CaC{{O}_{3}}$
Thus,
Hardness of $CaC{{l}_{2}}$ = 123 ppm
Therefore, hardness of effluent sample = $\dfrac{123\times 6}{28}=26ppm$

Hence, option (B) is correct.

Note:
Take a note that we have a proportion of quantities of drops of standard soap solution to determine the hardness of the sample. Without including those proportions do not calculate the hardness. As this step too will give an answer which would be one of the options but, unfortunately that would be wrong.