3.1 Seawater Chemistry

Step 1: Thermodynamic Basis for Equilibrium Constants

Each dissociation equilibrium is governed by the standard Gibbs free energy change. For the first dissociation of carbonic acid, CO₂(aq) + H₂O ⇌ H⁺ + HCO₃⁻, the equilibrium constant is related to the free energy by:

$$\Delta G^0 = -RT \ln K_1$$

Step 2: Writing K₁ in Terms of Activities

The thermodynamic equilibrium constant is expressed as a ratio of activities. Since we combine dissolved CO₂ and H₂CO₃ into CO₂* (because [H₂CO₃] is only ~0.3% of [CO₂(aq)]), we write:

$$K_1 = \frac{a_{\text{H}^+} \cdot a_{\text{HCO}_3^-}}{a_{\text{CO}_2^*}} = \frac{[\text{H}^+]\gamma_{\text{H}^+} \cdot [\text{HCO}_3^-]\gamma_{\text{HCO}_3^-}}{[\text{CO}_2^*]\gamma_{\text{CO}_2^*}}$$

Step 3: Stoichiometric (Apparent) Constants

In seawater, activity coefficients are folded into stoichiometric constants $K_1^*$ that use concentrations directly. This is valid at fixed ionic strength (salinity):

$$K_1^* = K_1 \cdot \frac{\gamma_{\text{CO}_2^*}}{\gamma_{\text{H}^+}\gamma_{\text{HCO}_3^-}} = \frac{[\text{H}^+][\text{HCO}_3^-]}{[\text{CO}_2^*]}$$

Step 4: Second Dissociation K₂

The second dissociation HCO₃⁻ ⇌ H⁺ + CO₃²⁻ follows identically. Writing the stoichiometric constant:

$$K_2^* = \frac{[\text{H}^+][\text{CO}_3^{2-}]}{[\text{HCO}_3^-]}$$

$K_2^*$ is roughly three orders of magnitude smaller than $K_1^*$, reflecting the greater difficulty of removing a proton from the already negatively charged bicarbonate ion.

Step 5: Temperature Dependence via van't Hoff

The temperature dependence of each equilibrium constant follows from the van't Hoff equation. For $K_1$, integrating gives the empirical Lueker et al. (2000) parameterization:

$$\frac{d \ln K}{dT} = \frac{\Delta H^0}{RT^2} \quad \Longrightarrow \quad \text{p}K_1 = \frac{3633.86}{T_K} - 61.2172 + 9.6777 \ln T_K - 0.011555\,S + 0.0001152\,S^2$$

Step 6: Solubility Product K_sp for CaCO₃

The dissolution equilibrium CaCO₃(s) ⇌ Ca²⁺ + CO₃²⁻ has a solubility product derived from the lattice energy and hydration energies of the ions. The thermodynamic $K_{sp}$ is:

$$K_{sp} = a_{\text{Ca}^{2+}} \cdot a_{\text{CO}_3^{2-}} = [\text{Ca}^{2+}]\gamma_{\text{Ca}^{2+}} \cdot [\text{CO}_3^{2-}]\gamma_{\text{CO}_3^{2-}}$$

Aragonite has a larger $K_{sp}$ ($\approx 6.65 \times 10^{-7}$) than calcite ($\approx 4.27 \times 10^{-7}$) because its orthorhombic crystal structure is less stable than the trigonal calcite structure. Pressure increases $K_{sp}$ due to the positive $\Delta V$ of dissolution.

Step 7: Pressure Dependence of K_sp

At depth, the increased pressure shifts $K_{sp}$ according to:

$$\ln\frac{K_{sp}(P)}{K_{sp}(0)} = -\frac{\Delta \bar{V}}{RT}(P - 1) + \frac{\Delta \bar{\kappa}}{2RT}(P - 1)^2$$

where $\Delta \bar{V}$ is the partial molar volume change and $\Delta \bar{\kappa}$ is the compressibility change. This causes $K_{sp}$ to increase with depth, creating the saturation horizon where $\Omega = 1$.

Step 1: Defining DIC as a Mass Balance

DIC is a conservative quantity (unaffected by proton transfer reactions). It represents the total dissolved inorganic carbon in all speciation forms:

$$\text{DIC} = [\text{CO}_2^*] + [\text{HCO}_3^-] + [\text{CO}_3^{2-}]$$

Step 2: Expressing Each Species in Terms of [H⁺] and DIC

Using $K_1$ and $K_2$, each species can be written as a fraction of DIC. Define the denominator $D = [\text{H}^+]^2 + K_1[\text{H}^+] + K_1 K_2$. Then:

$$[\text{CO}_2^*] = \frac{[\text{H}^+]^2}{D}\,\text{DIC}, \quad [\text{HCO}_3^-] = \frac{K_1[\text{H}^+]}{D}\,\text{DIC}, \quad [\text{CO}_3^{2-}] = \frac{K_1 K_2}{D}\,\text{DIC}$$

Step 3: Defining Total Alkalinity (TA) from Proton Balance

TA is defined as the excess of proton acceptors over proton donors relative to a reference species (CO₂*, H₂O, B(OH)₃). This yields the explicit alkalinity expression:

$$\text{TA} = [\text{HCO}_3^-] + 2[\text{CO}_3^{2-}] + [\text{B(OH)}_4^-] + [\text{OH}^-] - [\text{H}^+] + \text{minor bases}$$

Step 4: The Carbonate Alkalinity Approximation

At ocean pH (~8.1), the borate, hydroxide, and hydrogen ion contributions are relatively small (~3-5% of TA). Defining carbonate alkalinity $A_C$:

$$A_C = [\text{HCO}_3^-] + 2[\text{CO}_3^{2-}] \approx \text{TA} - [\text{B(OH)}_4^-] - [\text{OH}^-] + [\text{H}^+]$$

Step 5: Substituting Species Fractions into TA

Substituting the expressions from Step 2 into the carbonate alkalinity and using$[\text{B(OH)}_4^-] = B_T K_B / (K_B + [\text{H}^+])$:

$$\text{TA} = \text{DIC}\frac{K_1[\text{H}^+] + 2K_1 K_2}{[\text{H}^+]^2 + K_1[\text{H}^+] + K_1 K_2} + \frac{B_T K_B}{K_B + [\text{H}^+]} + \frac{K_w}{[\text{H}^+]} - [\text{H}^+]$$

This is an implicit equation in [H⁺]. Given any two of (TA, DIC, pH, pCO₂), the remaining two can be computed by solving this transcendental equation numerically (e.g., Newton-Raphson or bisection).

Step 6: Solving for pH from TA and DIC

Rearranging Step 5 as a residual function $f([\text{H}^+]) = \text{TA}_{\text{calc}} - \text{TA}_{\text{measured}} = 0$and applying Newton's method:

$$[\text{H}^+]_{n+1} = [\text{H}^+]_n - \frac{f([\text{H}^+]_n)}{f'([\text{H}^+]_n)}$$

Convergence is typically achieved in 3-5 iterations from an initial guess of $\text{pH} = 8.0$. The derivative $f'$ involves differentiating each species fraction with respect to [H⁺].

Step 7: Computing pCO₂ from the Solved System

Once [H⁺] is known, the dissolved CO₂ concentration and hence pCO₂ follow directly:

$$p\text{CO}_2 = \frac{[\text{CO}_2^*]}{K_0} = \frac{\text{DIC}}{K_0} \cdot \frac{[\text{H}^+]^2}{[\text{H}^+]^2 + K_1[\text{H}^+] + K_1 K_2}$$

Step 1: Definition of the Revelle Factor

The Revelle factor $R$ measures the fractional change in pCO₂ relative to the fractional change in DIC, at constant temperature, salinity, and total alkalinity:

$$R = \left(\frac{\partial \ln p\text{CO}_2}{\partial \ln \text{DIC}}\right)_{T,S,\text{TA}} = \frac{\text{DIC}}{p\text{CO}_2}\left(\frac{\partial p\text{CO}_2}{\partial \text{DIC}}\right)_{\text{TA}}$$

Step 2: Relating pCO₂ to DIC and [H⁺]

Since $p\text{CO}_2 = [\text{CO}_2^*]/K_0$, and [CO₂*] depends on both DIC and [H⁺], we need the chain rule. Using $[\text{CO}_2^*] = \text{DIC} \cdot \alpha_0([\text{H}^+])$:

$$\frac{d\,p\text{CO}_2}{d\,\text{DIC}} = \frac{1}{K_0}\left(\alpha_0 + \text{DIC}\frac{\partial \alpha_0}{\partial [\text{H}^+]}\frac{\partial [\text{H}^+]}{\partial \text{DIC}}\right)$$

Step 3: Implicit Differentiation of the TA Constraint

Since TA is held constant, differentiating the TA expression with respect to DIC yields$\partial[\text{H}^+]/\partial\text{DIC}$ implicitly:

$$0 = \frac{\partial \text{TA}}{\partial \text{DIC}}\bigg|_{[\text{H}^+]} + \frac{\partial \text{TA}}{\partial [\text{H}^+]}\bigg|_{\text{DIC}} \cdot \frac{\partial [\text{H}^+]}{\partial \text{DIC}} \quad \Longrightarrow \quad \frac{\partial [\text{H}^+]}{\partial \text{DIC}} = -\frac{\partial \text{TA}/\partial \text{DIC}}{\partial \text{TA}/\partial [\text{H}^+]}$$

Step 4: Evaluating the Partial Derivatives

The partial derivative of TA with respect to DIC at constant [H⁺] involves only the ionization fractions:

$$\frac{\partial \text{TA}}{\partial \text{DIC}}\bigg|_{[\text{H}^+]} = \alpha_1 + 2\alpha_2 = \frac{K_1[\text{H}^+] + 2K_1 K_2}{[\text{H}^+]^2 + K_1[\text{H}^+] + K_1 K_2}$$

The derivative $\partial\text{TA}/\partial[\text{H}^+]$ is more complex, involving the derivatives of all species fractions and the borate term with respect to [H⁺].

Step 5: Simplified Analytical Expression

Neglecting the borate and water terms, the Revelle factor can be expressed analytically in terms of the carbonate alkalinity $A_C$ and the species fractions:

$$R \approx \frac{\text{DIC} \cdot A_C}{[\text{CO}_3^{2-}](2\,\text{DIC} - A_C)} = \frac{\text{DIC}}{[\text{CO}_3^{2-}]} \cdot \frac{A_C}{2\,\text{DIC} - A_C}$$

This shows that $R$ increases as [CO₃²⁻] decreases (i.e., as more CO₂ is absorbed), confirming the positive feedback: ocean buffering weakens as DIC rises. Typical present-day values are $R \approx 10$; at doubled CO₂ levels, $R$ could reach 15-17.