What a Piper diagram represents
A Piper diagram represents the relative ionic composition of a water sample. Concentrations are first converted to milliequivalents per litre, because an equivalent expresses charge rather than mass. Each triangle closes independently to 100%: Ca²⁺, Mg²⁺ and Na⁺+K⁺ among the cations; HCO₃⁻+CO₃²⁻, SO₄²⁻ and Cl⁻+NO₃⁻ among the anions. The resulting point preserves proportions, not absolute concentration.
ci(meq/L)=ci(mg/L)Mi∣zi∣ pi=100∑j∈same signcj(meq/L)ci(meq/L) Rays projected from both triangles intersect in the diamond. Its natural coordinates are u=%(Na+K) among the cations and v=%(SO4+Cl+NO3) among the anions, so a single point combines the alkaline-earth–alkali and weak-acid–strong-acid relations. Because each triangle closes independently, an imbalance between cations and anions does not displace the point: it is recorded separately, in the charge-balance error.
Nitrate in the anion triangle
Piper lists nitrate among the strong-acid radicals and adds second-rank constituents to the group they are chemically related to. NO₃⁻ is therefore grouped with Cl⁻ and takes part in the anion triangle, the diamond, the Palmer properties, the charge balance, the ionic strength and the analysed-ion sum.
Palmer properties and subareas 5–9
Piper interprets the diamond through the four properties defined by Palmer. Let A be the alkaline earths, B the alkalis, W the weak acids and S the strong acids, all as equivalent percentages:
Alkalinity
SA=min(A,W)PA=max(0,W−A) Salinity
SS=max(0,A−W)PS=min(B,S) SA, SS, PS and PA are secondary alkalinity, secondary salinity, primary salinity and primary alkalinity, and they add to 100%. They describe the balance between ionic groups; they do not imply that a particular molecular salt is present in solution. Each subarea of the diamond corresponds to the region where one of those properties exceeds 50%.
| Area | Property | Chemical criterion |
|---|
| 5 | Secondary alkalinity | Ca²⁺+Mg²⁺ exceed SO₄²⁻+Cl⁻+NO₃⁻; alkaline-earth alkalinity remains. |
| 6 | Secondary salinity | SO₄²⁻+Cl⁻+NO₃⁻ exceed Na⁺+K⁺; alkaline-earth salinity remains. |
| 7 | Primary salinity | Na⁺+K⁺ and strong-acid anions each exceed 50%. |
| 8 | Primary alkalinity | Na⁺+K⁺ exceed SO₄²⁻+Cl⁻+NO₃⁻; alkali alkalinity remains. |
| 9 | No dominant pair | None of the preceding pairs reaches an exclusive property. |
Charge balance and derived quantities
CBE(%)=100∑C+∑A∑C−∑A The charge-balance error compares the equivalent sum of cations with that of anions. A value near zero indicates agreement between the measured charges; it carries no information about the accuracy of the analysis, and a systematic error affecting cations and anions equally cancels out. For major-ion analyses, the hydrogeological literature takes |5|% as the usual acceptance margin.
H(mg/L as CaCO3)=2.497[Ca]+4.118[Mg] I=21i∑mizi2=20001i∑ci(meq/L)∣zi∣ Total hardness expresses the Ca and Mg content as an equivalent mass of CaCO₃. Ionic strength is computed over the nine entered ions and includes no unanalysed species, activity coefficients or complexation, so it corresponds to the concentration-based ionic strength.
Conversion factors
The equivalent weight of an ion is its molar mass divided by the absolute value of its charge; multiplying mg/L by the inverse factor gives meq/L. The values come from the IUPAC standard atomic weights and reproduce table 9 of Hem (1985) to four significant figures.
| Ion | Equivalent weight | mg/L → meq/L |
|---|
| Ca²⁺ | 20.03900 | 0.04990269 |
| Mg²⁺ | 12.15250 | 0.08228760 |
| Na⁺ | 22.99000 | 0.04349717 |
| K⁺ | 39.09800 | 0.02557676 |
| HCO₃⁻ | 61.01600 | 0.01638914 |
| CO₃²⁻ | 30.00400 | 0.03332889 |
| SO₄²⁻ | 48.02800 | 0.02082119 |
| Cl⁻ | 35.45000 | 0.02820874 |
| NO₃⁻ | 62.00400 | 0.01612799 |