In what follows, I analyze how much the warming of the surface waters of the ocean would be expected to increase the concentration of CO2 in the atmosphere.
Table of Contents
- Setup
- Key Relationships
- Changing to Logarithmic Variables
- Implication of Revelle Factor
- Derivation
- Result
Setup
Variables
This analysis makes use of the following state variables:
― Concentration of gaseous CO2 as a molar fraction of air
― Concentration of aqueous CO2 as a molar fraction of seawater solution
― Concentration of Dissolved Inorganic Carbon (DIC) as a molar fraction of seawater solution
― Total moles of air molecules
― Total moles of molecules in seawater layer subject to warming
― Total moles of CO2 in air
― Total moles of carbon in seawater layer subject to warming
― Absolute temperature of seawater layer subject to warming
Parameters
The following parameters will be referenced:
― Dimensionless Henry’s Law volatility ratio for CO2
― Enthalpy of dissolution of CO2 in seawater
― Gas constant
― Revelle factor (aka buffering factor) for CO2 in seawater layer subject to warming
Assumptions
This analysis relies on the following assumptions:
- There is a defined layer of seawater of temperature
in contact with the atmosphere which undergoes a temperature change
.
- The net amount of carbon which leaves the layer of warmed seawater becomes a net increase in gaseous CO2 in the atmosphere.
- The layer of seawater subject to warming is uniform in temperature and has uniform concentrations of CO2(aq) and DIC.
- The atmosphere has a uniform concentration of gaseous CO2.
- Sufficient time has passed to allow the warmed layer of seawater to come into full CO2 gas-exchange equilibrium with the atmosphere.
This analysis will compute the change in atmospheric CO2 concentration, , which results from the layer of seawater being warmed by a temperature change
which is small in comparison to the absolute temperature
.
Logarithm Notation
This analysis makes heavy use of logarithms of values, since key formulas such as the Van ‘t Hoff equation for temperature change in Henry’s ratio, and the Revelle factor, are most naturally expressed in terms of logarithms.
To simplify the use of logarithms, the notation will be use to denote the natural log of
, i.e.,
.
Key Relationships
Concentrations and Total Carbon
The total carbon in the atmosphere and the seawater layer subject to warming are given, respectively, by:
(1)
(2)
Conservation of Carbon
Because net carbon leaving the warming layer of seawater is assumed to become CO2 in the atmosphere, the following relationship must hold.
(3)
(4)
Henry’s Law
Henry’s Law expresses how the concentration of a molecular species in the gaseous phase relates to the concentration of that molecular species in its dissolved aqueous form. Henry’s Law is given by:1The formula used for Henry’s Law here uses Henry’s volatility ratio, . Alternatively, we could have used Henry’s solubility ratio
, in which case the equation would have been:
. Either formulation leads to the same end results.
(5)
Van ‘t Hoff equation
The Van ‘t Hoff equation describes how an “equilibrium constant” changes with temperature. It is applicable to the Henry’s Law ratio. It tells us that the logarithm of Henrys ratio, , varies with temperature as follows:2Note that the sign of the right-hand-side of the equation would be opposite, if one used Henry’s solubility ratio
, instead of Henry’s volatility ratio,
.
(6)
where
(7)
The symbol is non-standard, and is used to make the analysis somewhat less verbose. For most gases (including CO2), the enthalpy of dissolution,
, is negative, so
is positive. Note that
has units of Kelvin. For CO2,
has a value of 2400 K.
Revelle Factor
The Revelle factor characterizes how buffering in seawater affects shifts in the concentration of CO2(aq) versus the total concentration of dissolved inorganic carbon (DIC):3“Inorganic carbon”, in this context, involves molecules which contain carbon but which do not have both a C-H and a C-C bond.
(8)
For seawater, is typically in the range 8-13.
Changing to Logarithmic Variables
Some of the principles were not expressed in terms of logarithms of concentrations. This section finds ways to express those formulas in terms of logarithms of concentrations.
Derivatives of Logarithms of Concentrations
(9)
(10)
Conservation of Carbon in Terms of Logarithmic Concentrations
By combining the formula for conservation of carbon with the formulas for derivatives of the logarithms of concentrations, one obtains:
(11)
(12)
(13)
This last equation expresses the formula for conservation of carbon in terms of the logarithms of concentrations.
Logarithmic Version of Henry’s Law
Taking the logarithm of the Henry’s Law formula yields:
(14)
Implication of Revelle Factor
The Revelle factor may be used to relate changes in and
as follows:
(15)
(16)
Derivation
We begin by taking the derivative of the logarithmic version of Henry’s Law:
(17)
Next, we apply the Van ‘t Hoff equation:
(18)
Applying the implication of the Revelle Factor leads to:
(19)
Then, we apply conservation of carbon:
(20)
Rearrangement leads to:
(21)
Finally, expressing the result in terms of concentration rather than log of concentration yields:
(22)
Result
For a small temperature change , the preceding formula for the derivative leads to:
(23)
(24)
- 1The formula used for Henry’s Law here uses Henry’s volatility ratio,
. Alternatively, we could have used Henry’s solubility ratio
, in which case the equation would have been:
. Either formulation leads to the same end results.
- 2Note that the sign of the right-hand-side of the equation would be opposite, if one used Henry’s solubility ratio
, instead of Henry’s volatility ratio,
.
- 3“Inorganic carbon”, in this context, involves molecules which contain carbon but which do not have both a C-H and a C-C bond.