Saruhashi: Fractions and Total Amount

Read the selected approximation

The paper treats activity coefficients as approximately unity for an extremely dilute fresh-water model. Its two dissociation relations connect H₂CO₃, HCO₃⁻ and CO₃²⁻; its H₂CO₃ convention includes dissolved free CO₂.

This is the specific approximation used below. It is not a general assertion that all natural waters behave ideally or that the same constants can be carried over to sea water.

Sources for this section: [1]

Original ratio chain

Let F denote the paper’s combined H₂CO₃ group, B bicarbonate and C carbonate. H is hydrogen-ion concentration; K₁ and K₂ are dissociation constants. In the invented ideal dilute model, use consistent concentration units and stipulate positive H with ratios K₁/H=10 and K₂/H=0.001. The relations give B/F=10 and C/B=0.001. Find F: B: C and fractions that sum to 1.

Supply total concentration T=2 mmol/L and calculate each concentration. Repeat for T=4 mmol/L while retaining the stipulated ratios. No actual pH, temperature or sampled water is reconstructed; the ratios and totals are original numerical inputs.

Sources for this section: [1]

Reference answer: the ratios form a chain

Taking F=1 gives B=10 and C=0.01, so F: B: C=1: 10: 0.01. The sum is 11.01. The fractions are 100/1101,1000/1101 and 1/1101, approximately 0.0908265,0.9082652 and 0.0009083. They are nonnegative and sum exactly to 1.

At T=2 mmol/L, the concentrations are 200/1101,2000/1101 and 2/1101 mmol/L, approximately 0.18165,1.81653 and 0.0018165. At T=4 they double. The fractions do not change because the two ratios were held fixed.

A common error is to use C/F=0.001. The supplied second ratio is C/B, so C/F=(C/B)(B/F)=0.01. Another is to present the fractions as absolute concentrations without supplying a total. These are different errors: one misreads a ratio, the other omits a scale.

Sources for this section: [1]

Return to the purpose of the tables

The historical tables were intended to reduce repeated calculations from total amount, pH and temperature, with separate treatment of electrolytes in sea water.

The original fixture makes one part of that logic visible. It does not reproduce the tables, provide a contemporary seawater model or infer how an actual ocean has changed. The constants and the total must each be appropriate to the question being asked.

Sources for this section: [1]