A sample of hard water contains \( 120\text{ mg} \) of \( \text{MgSO}_4 \) (molar mass \( 120\text{ g mol}^{-1} \)) per litre. What is the concentration of magnesium ions expressed in parts per million (ppm) by mass, assuming the density of water is \( 1.0\text{ g mL}^{-1} \)?

Mass of \( \text{MgSO}_4 \) in \( 10^6\text{ mg} \) of water is \( 120\text{ mg} \). Moles of \( \text{Mg}^{2+} \) = \( 120\text{ mg} / 120\text{ g mol}^{-1} = 1\text{ mmol} \). Mass of \( \text{Mg}^{2+} \) = \( 1\text{ mmol} \times 24\text{ mg mmol}^{-1} = 24\text{ mg} \) per \( 10^6\text{ mg} \), giving \( 24\text{ ppm} \) if considering whole salt, but for \( \text{Mg}^{2+} \) alone: \( (24 / 120) \times 120 = 24\text{ mg/L} \implies 24\text{ ppm} \) total salt, wait: \( \text{Mg} \) mass fraction in \( \text{MgSO}_4 \) is \( 24 / 120 = 0.2 \). Thus \( 120 \times 0.2 = 24\text{ mg/L} \equiv 24\text{ ppm} \). Let's recheck: \( 120\text{ mg/L} = 120\text{ mg} / 10^6\text{ mg} = 120\text{ ppm} \) of salt, and for \( \text{Mg}^{2+} \) it is \( 120 \times (24/120) = 24\text{ ppm} \).