GATE 2025 CE (CE1) – Question 44
$MgCl_2$ and $CaSO_4$ salts are added to 1 litre of distilled deionized water and mixed until completely dissolved. Total Dissolved Solids (TDS) concentration is 500 mg/l, and Total Hardness (TH) is 400 mg/l (as $CaCO_3$). The amounts of $MgCl_2$ and $CaSO_4$ added are calculated (rounded off to the nearest integer). Which of the following options is/are true:
Atomic weights: Ca(40), Mg(24), S(32), O(16), Cl(35.5), C(12)
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Correct answer: (C) Amount of MgCl$_2$ added is 103 mg; (D) Amount of CaSO$_4$ added is 397 mg
Explanation
The molar masses are 95 for $MgCl_2$, 136 for $CaSO_4$ and 100 for $CaCO_3$. Let the amounts be $m_1$ and $m_2$ mg. The total solids give $m_1 + m_2 = 500$, and the hardness, as $CaCO_3$, gives $\frac{100}{95}m_1 + \frac{100}{136}m_2 = 400$. Solving gives $m_1 \approx 102$ mg and $m_2 \approx 398$ mg, so the amounts are $MgCl_2$ about 102 mg and $CaSO_4$ about 398 mg. The two options that agree with this to within a milligram are 103 mg and 397 mg (they add up to 500 and give a hardness of 400 mg/l as $CaCO_3$ to the nearest unit), so C and D are the intended answers, while 143 mg and 357 mg would give a hardness of about 413 mg/l.