3.2 Dissolved Gases

Step 1: Chemical Potential at Equilibrium

At air-sea equilibrium, the chemical potential of a gas must be equal in both phases. For gas species $i$ partitioned between the gas phase (g) and dissolved phase (aq):

$$\mu_i^{(\text{g})} = \mu_i^{(\text{aq})}$$

Step 2: Expressing Chemical Potentials

For an ideal gas, $\mu_i^{(\text{g})} = \mu_i^{0,\text{g}} + RT \ln p_i$. For the dissolved species, using the molality scale with activity coefficient $\gamma_i$:

$$\mu_i^{0,\text{g}} + RT \ln p_i = \mu_i^{0,\text{aq}} + RT \ln(\gamma_i m_i)$$

Step 3: Isolating the Concentration-Pressure Relationship

Rearranging and defining the standard free energy of dissolution$\Delta G_{\text{sol}}^0 = \mu_i^{0,\text{aq}} - \mu_i^{0,\text{g}}$:

$$RT \ln\frac{p_i}{\gamma_i m_i} = \Delta G_{\text{sol}}^0 \quad \Longrightarrow \quad \frac{p_i}{\gamma_i m_i} = \exp\left(\frac{\Delta G_{\text{sol}}^0}{RT}\right) \equiv K_H$$

Step 4: Henry's Law in the Dilute Limit

For dilute solutions where $\gamma_i \to 1$, this reduces to the familiar form. The solubility coefficient $K_0 = 1/K_H$ gives concentration per unit pressure:

$$p_i = K_H \cdot C_i \qquad \text{or equivalently} \qquad C_{\text{sat}} = K_0 \cdot p_i$$

Step 5: Temperature Dependence (van't Hoff)

The temperature dependence of $K_H$ follows from the enthalpy of dissolution$\Delta H_{\text{sol}}$. Integrating the van't Hoff equation between reference temperature $T^*$ and $T$:

$$\ln\frac{K_H(T)}{K_H(T^*)} = -\frac{\Delta H_{\text{sol}}}{R}\left(\frac{1}{T} - \frac{1}{T^*}\right)$$

Since dissolution of gases is exothermic ($\Delta H_{\text{sol}} < 0$),$K_H$ increases (solubility decreases) with temperature, explaining why cold water holds more gas.

Step 6: Salting-Out Effect (Setchenov Equation)

In electrolyte solutions, ions disrupt the water structure and reduce the solubility of neutral gas molecules. The Setchenov (Sechenov) equation accounts for this:

$$\ln\frac{K_H(S)}{K_H(0)} = k_s \cdot S \quad \Longrightarrow \quad \ln\gamma_{\text{gas}} = k_s \cdot S$$

where $k_s$ is the Setchenov coefficient (positive for salting-out). For O₂ in seawater, $k_s \approx 0.006$ per PSU, reducing solubility by ~20% at S = 35.

Step 1: The Thin-Film (Stagnant Boundary Layer) Model

Near the air-sea interface, turbulent mixing is suppressed, and gas transport occurs by molecular diffusion across a thin boundary layer of thickness $\delta$. Fick's first law gives the flux:

$$F = D \frac{C_{\text{sat}} - C_{\text{bulk}}}{\delta} = k(C_{\text{sat}} - C)$$

where the piston velocity (gas transfer velocity) is identified as $k = D/\delta$, with $D$ the molecular diffusivity of the gas in water.

Step 2: Schmidt Number Scaling

The boundary layer thickness depends on the ratio of momentum diffusivity (kinematic viscosity $\nu$) to mass diffusivity ($D$). The Schmidt number captures this ratio:

$$Sc = \frac{\nu}{D}$$

Boundary layer theory predicts $\delta \propto Sc^n$, where $n = 1/2$ for a smooth surface (surface renewal model) or $n = 2/3$ for a rigid surface (film model).

Step 3: Wind Speed Dependence

Wind stress generates turbulence at the surface, thinning the boundary layer. Empirically,$k$ scales as $u_{10}^2$ at moderate to high wind speeds (consistent with a drag-law scaling where stress $\tau \propto u_{10}^2$). Combining:

$$k = a \cdot u_{10}^2 \cdot \left(\frac{Sc}{660}\right)^{-1/2}$$

Normalized to $Sc = 660$ (CO₂ at 20 degrees C). Wanninkhof (1992) determined $a = 0.31$using global bomb-¹⁴C inventories as a constraint on the globally averaged transfer velocity.

Step 4: Surface Renewal Model

An alternative to the thin-film model: turbulent eddies periodically bring fresh bulk water to the surface, where it equilibrates by diffusion for a characteristic time $\tau$before being replaced. The resulting transfer velocity is:

$$k = \sqrt{\frac{D}{\pi \tau}} \propto D^{1/2} \propto Sc^{-1/2}$$

This model naturally gives the $Sc^{-1/2}$ dependence observed in wind-tunnel experiments, supporting the Wanninkhof parameterization.

Step 5: Bubble-Mediated Transfer at High Winds

At wind speeds above ~10 m/s, breaking waves inject bubbles below the surface. Gas dissolves from pressurized bubbles, adding a flux component that is nearly independent of solubility:

$$F_{\text{total}} = k(C_{\text{sat}} - C) + F_{\text{bubble}} \quad \text{where} \quad F_{\text{bubble}} = b \cdot u_{10}^3 \cdot p_{\text{gas}} \cdot V_b$$

The cubic wind speed dependence for bubble injection enhances the transfer of less soluble gases (e.g., O₂, N₂) more than soluble ones (CO₂), creating supersaturation of N₂ and O₂ in storm conditions.

Step 1: Starting from Henry's Law for O₂

The saturation concentration of O₂ in seawater at 1 atm total pressure is determined by the atmospheric O₂ mole fraction ($x_{\text{O}_2} = 0.20946$) and the Henry's law solubility:

$$C_{\text{sat}}^{\text{O}_2} = K_0^{\text{O}_2}(T, S) \cdot (P_{\text{atm}} - p_{\text{H}_2\text{O}}) \cdot x_{\text{O}_2}$$

where $p_{\text{H}_2\text{O}}$ is the water vapor pressure (subtracted because only dry gas contributes).

Step 2: Empirical Parameterization of K₀

Garcia and Gordon (1992) fit the solubility to a polynomial in the scaled temperature variable$T_s$, which compresses the temperature range for better polynomial behavior:

$$T_s = \ln\left(\frac{298.15 - T}{273.15 + T}\right)$$

This transformation maps the oceanographic temperature range (roughly -2 to 40 degrees C) onto a roughly linear scale, since $T_s$ is monotonically decreasing with temperature.

Step 3: Full Garcia-Gordon Formula

The natural log of the saturation concentration is expanded as a polynomial in $T_s$with salinity-dependent corrections:

$$\ln C_{\text{sat}} = A_0 + A_1 T_s + A_2 T_s^2 + A_3 T_s^3 + A_4 T_s^4 + A_5 T_s^5 + S(B_0 + B_1 T_s + B_2 T_s^2 + B_3 T_s^3) + C_0 S^2$$

Step 4: Physical Origin of Each Term

The polynomial structure arises from expanding the thermodynamic functions:

$$\ln K_0 = -\frac{\Delta G_{\text{sol}}^0}{RT} = -\frac{\Delta H_{\text{sol}}^0}{RT} + \frac{\Delta S_{\text{sol}}^0}{R} + \frac{\Delta C_p}{R}\left(\frac{T^*}{T} - 1 + \ln\frac{T}{T^*}\right) + \cdots$$

The leading terms ($A_0, A_1$) capture the enthalpy and entropy of dissolution. Higher-order terms ($A_2$ onward) account for the heat capacity change upon dissolution. The salinity terms ($B_i$) represent the salting-out effect from the Setchenov relation.

Step 5: Pressure Correction at Depth

For in-situ saturation at depth, the hydrostatic pressure increases the solubility via the partial molar volume of O₂ in solution $\bar{V}_{\text{O}_2}$:

$$\ln\frac{K_0(P)}{K_0(1\,\text{atm})} = \frac{\bar{V}_{\text{O}_2}(P - 1)}{RT}$$

With $\bar{V}_{\text{O}_2} \approx 32$ cm³/mol, the pressure correction increases solubility by roughly 1.4% per 1000 dbar, a small effect relevant mainly for deep water calculations.