 Solution

Fresh rainwater is ordinarily soft because evaporation leaves most nonvolatile dissolved salts behind; “always” is an idealization, not an exceptionless law. Water hardness is principally caused by dissolved calcium and magnesium ions. Water vapour rising from a hard-water source does not carry these ions with it as vapour. Condensation consequently supplies low-mineral water before contact with dust, surfaces or soil. Mineral particles, sea spray and a collecting surface can add salts, so actual collected rainwater need not have zero water hardness. Nor does softness establish purity or freedom from microorganisms.^[ref]

The historically appropriate quantitative principle is soap titration of water hardness. Soluble soap reacts with hardness-producing ions to form poorly soluble salts, using up soap before a persistent lather can form. For a representative carboxylate soap,Magnesium behaves similarly. Once the dissolved hardness-producing ions have substantially reacted, additional soap can sustain the specified lather. The endpoint is therefore an operational observation under standardized shaking, standing time, temperature and sample volume; it is not the first transient bubble.

Use a measured volume of sample in a stoppered vessel with enough headspace for shaking. Add a soap solution of known strength incrementally from a burette, shaking in a reproducible way, until the chosen persistent-lather endpoint appears. Determine a blank with a corresponding amount of soft or distilled water, because some soap is needed for the lather even when no hardness salts are present. Standardize the soap against a reference solution containing a known calcium amount, expressed as an equivalent mass of calcium carbonate. This standardization is essential: the apparent strength cannot safely be inferred from the mass of an incompletely characterized commercial soap.

Let $V_s$ be the sample volume in millilitres, $V$ the soap titre in millilitres and $B$ the corresponding blank. If $f$ is the standardized capacity in milligrams of calcium carbonate equivalent per millilitre of soap solution, thenFor a standard containing $m_0$ milligrams of calcium carbonate equivalent and requiring blank-corrected titre $V_0-B_0$, $f=m_0/(V_0-B_0)$. This makes the measurement quantitative without assuming an unreported soap concentration or inventing a sample titre. Historical results could instead be stated as grains of calcium carbonate equivalent per gallon, provided the gallon convention was specified.

Repeating the examination after controlled boiling, cooling, removal of the precipitate and restoration of the original volume estimates the residual or permanent hardness. The reduction estimates the boiling-removable fraction, often called temporary hardness. A representative reaction isCarbonate equilibria and magnesium chemistry make this more subtle than assuming every dissolved salt is destroyed by boiling. Very hard samples may need measured dilution; magnesium, other metal ions and contamination can disturb the lather endpoint. Thus the soap result is an experimentally calibrated hardness equivalent, not an exact identification of every original salt.