# Calculation from Equilibrium Equations

The relevant equilibrium equations are as follows:

$$
H_2O \rightleftharpoons H^+ + OH^-
$$
(6a)

$$
K_w = \{H^+\}\{OH^-\} = 10^{-14} \text{ (at 25°C)}
$$
(6b)

$$
H_2CO_3 \rightleftharpoons H^+ + HCO_3^-
$$
(7a)

$$
K_{A1} = \frac{\{H^+\}\{HCO_3^-\} }{\{H_2CO_3\}} = 10^{-6.3} \text{ (at 25°C)}
$$
(7b)

$$
HCO_3^- \rightleftharpoons H^+ + CO_3^{2-}
$$
(8a)

$$
K_{A2} = \frac{\{H^+\}\{CO_3^{2-}\} }{\{HCO_3^-\}} = 10^{-10.3} \text{ (at 25°C)}
$$
(8b)

The alkalinity of water can be expressed as follows:

$$
\text{Total alkalinity (mg/l as } CaCO_3) / 50,000 = (HCO_3^-) + 2(CO_3^{2-}) + (OH^-) - (H^+)
$$
(9)

From the above equations, the following expressions for different alkalinities can be derived:

**Carbonate alkalinity (mg/l as CaCO3)**
$$
\text{Carbonate alkalinity (mg/l as } CaCO_3) = \frac{50,000 \{(\text{alkalinity}/50,000) + (H^+) - (K_w / (H^+))\}}{1 + \{(H^+)/2K_{A2}\}}
$$
(10)

**Bicarbonate alkalinity (mg/l as CaCO3)**
$$
\text{Bicarbonate alkalinity (mg/l as } CaCO_3) = \frac{50,000 \{(\text{alkalinity}/50,000) + (H^+) - (K_w / (H^+))\}}{1 + \{2K_{A2}/(H^+)\}}
$$
(11)

It should be noted that the equilibrium constants vary with temperature and activities of ions vary with concentration of ions in water. If temperature of water varies significantly from 25 °C and ionic strength of water is high (that is high concentration dissolved ions), appropriate corrections need to be incorporated in the above equations. Otherwise these equations would give reasonably accurate results. Sometimes actual concentrations of carbonate and bicarbonate ions may be desired. Conversions to mg/l of CO32- and HCO3- are as follows:

$$
\text{mg/l } CO_3^{2-} = \text{mg/l carbonate alkalinity} \times 0.6
$$
(12)

$$
\text{mg/l } HCO_3^- = \text{mg/l bicarbonate alkalinity} \times 1.22
$$
(13)

Molar concentrations may be obtained as follows:

$$
(CO_3^{2-}) = \frac{(\text{mg/l } CO_3^{2-})}{60,000}
$$
(14)

$$
(HCO_3^-) = \frac{(\text{mg/l } HCO_3^-)}{61,000}
$$
(15)