# Calculation of f(CO_{2}) for Moist Air Conditions

The first step in evaluating the *in situ* gas concentration, either in the atmosphere or in vapor equilibrated with surface seawater, is calculation of the gas fugacity in moist air from the measured mole fraction in dry air. Weiss and Price (1980) give the theoretical basis for this calculation based on equations given by Guggenheim (1967, pp. 175-77) for calculating fugacities in binary mixtures:

f_{1} = x_{1}P exp[(B_{11} + 2(x_{2})^{2} · Δ_{12})P/RT] (eq. 18)

where x_{1} is the mole fraction of pure gas 1, x_{2} is the mole fraction of pure gas 2, P is the total pressure, R is the gas constant, T is the absolute temperature, and Δ_{12} is defined by:

B_{12} = 1/2(B_{11} + B_{22}) + Δ_{12} (eq. 19)

where B_{11} is the virial coefficient for interaction between pure gas 1 molecules; B_{22} is the virial coefficient for interaction between pure gas 2 molecules; and B_{12} is the virial coefficient for interaction between molecules of gases 1 and 2.

For calculating *in situ* gas fugacities, gas 1 is here considered as the analyte gas and gas 2 as dry air. The x_{1} in the above equation is the mole fraction of analyte gas in the analyte gas-dry air mixture. For an analyte like CO_{2}, the atmospheric value of x_{1} is approximately 350 · 10^{-6} moles of CO_{2} per mole of dry gas mixture. The x_{2} is the mole fraction of dry air in that same mixture, and is approximately equal to 1:

x_{2} = (1-350 · 10^{-6})/1.0 = 0.99965 moles of air per mole of dry gas mixture (eq. 20)

To calculate CO_{2} fugacity for the moist air conditions at the air-sea interface, the measured mole fraction of CO_{2} in dry air, x_{1} in equation 18 above, must be corrected to the mole fraction of the CO_{2} in moist air. If the air-sea interface is regarded as saturated in water vapor at the *in situ* temperature, the mole fraction of the CO_{2} in dry air, x_{1}, can be corrected to the mole fraction in moist air x_{1}´ as follows:

x_{1}´ = x_{1}(1-p_{sw}/p_{atm}) (eq. 21)

where p_{sw} is the saturated vapor pressure of seawater at the temperature of the measurement. For atmospheric f(CO_{2}), the temperature of calculation is the *in situ* air temperature; for f(CO_{2}) in surface seawater, this corresponds to the equilibrator temperature. The total barometric pressure is represented by p_{atm}.

Substituting x_{1}´ from equation 21 for x_{1} in equation 18 gives:

f_{1} = x_{1}(1-p_{sw}/p_{atm})P_{atm} exp[(B_{11} + 2(x_{2})^{2} · Δ_{12})p_{atm}/RT] (eq. 22)

Since x_{2} is approximately equal to 1 for the analyte gases considered here (equation 20), equation 22 reduces to:

f = x_{1}(p_{atm} - p_{sw}) exp[p_{atm}(B + 2Δ)/RT] (eq. 23)

where:

f is the fugacity of the analyte gas in moist air in units of atmospheres

x_{1} is the measured mole fraction of the analyte gas in dry air in units of parts per million (ppm)

p_{atm} is the total barometric pressure in units of atmospheres

p_{sw} is the saturated vapor pressure of seawater (in atmospheres) at the temperature of the measurements and is calculated from equation (3) in the main text from Weiss and Price(1980):

ln p_{sw} = 24.4543-67.4509(100/T)-4.8489 ln(T/100)-0.000544S (eq. 24)

B is the virial coefficient for CO_{2} and can be calculated using a power series given by Weiss(1974):

B=-1636.75 + 12.0408T - 3.27957 · 10^{-2}T^{2} + 3.16528 · 10^{-5}T^{3} (eq. 25)

Δ is the cross virial coefficient B_{12} for interaction between gases 1 and 2 minus the mean of B_{11} and B_{22} for two pure gases (see equation 19). Weiss (1974) gives this for CO_{2} and air as a function of temperature.

Δ = 57.7 - 0.118T cm^{3}/mole (eq. 26)

R is the gas constant

T is the temperature of water in the equilibrator in Kelvin at the time the gas aliquot was removed.

The fugacity obtained is the fugacity of CO_{2} in the moist equilibrator vapor. Since the temperature in the equilibrator is higher than the sea surface temperature, another calculation is required to correct this value to obtain the fugacity of CO_{2} at the *in situ* sea surface conditions.