Douce A.P. Thermodynamics of the Earth and Planets

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648 Thermodynamics of life

The partial derivative of the Gibbs free energy of oxygen follows equations (9.96) and

(9.100):

0,O

1,T

+lnf O

+2λ

=0 (14.6)

and the partial derivative for liquid water, ignoring the effect on pressure on the chemical

potential of liquid H

O, is (compare equation (9.101) for graphite):

0,H

(

liquid

)

1,T

+2λ

+λ

=0. (14.7)

Equations (14.2)–(14.7) constitute a system of 14 equations in the 14 unknowns:

O(liquid)

, n

O(vapor )

, n

, ln(fO

), P , λ

, λ

and

is given by (14.1)). Numerical solution with Maple is straightforward and is briefly

discussed in Software Box 14.1. Recall that the four Lagrange multipliers are necessary in

order to solve the constrained minimum problem (Section 9.6.2), but we have no use for

their numerical values.

Let us first examine solution sets at 25

◦

C for a planet with Earth’s gravitational accel-

eration and with total volatile content adjusted so as to yield an atmospheric mass of order

kg m

−2

, comparable to the present day Earth. We start with the following values.

=1650 ×10

mols m

−2

=650 ×10

mols m

−2

=350 ×10

mols m

−2

=100 ×10

mols m

−2

The total volatile mass that results is equal to 1.765×10

kg m

−2

, which is almost twice the

present-day terrestrial atmospheric mass, but as we shall see some of this mass condenses

as liquid water. The relative proportions of the four components are similar to cometary

material (see Lodders & Fegley, 1998, Tables 15.3 and 15.4), but I emphasize that there is

no special significance to this starting composition, other than being plausible. The point of

this exercise is to examine how atmospheric speciation responds to changes in the model

parameters.

Figure 14.1 shows the effect of changing the bulk contents of H, C and O independently,

i.e. the bulk content of one each of these elements changes, while all the others remain

constant and equal to the values listed above. The independent variable in each graph is the

bulk atomic fraction of the component that varies. Temperature is kept constant at 25

◦

and g = 9.8ms

−2

. The top panels show the oxygen fugacity and mol fraction of each of

the gas species except CO, which never rises above ∼ 10

−10

and which we discuss later.

The bottom panels display atmospheric pressure at the planet’s surface and the amount of

O that condenses as liquid, converted to thickness of the water column (55.56×10

mols

−2

=1 m of water depth).

The calculations show a steep jump in species distribution, between a H

–NH

–CH

atmosphere with virtually no CO

, and a CH

–CO

atmosphere with small but non-zero

contents of NH

and H

. The atmosphere becomes less reduced in response to decreasing

bulk H content or increasing bulk O or C contents, but in every case there is a sudden jump

in species abundances over very narrow bulk composition intervals. The behavior is in some

649 14.1 Chemical evolution of post-nebular atmospheres

–6

–5

–4

–3

–2

–1

–72

–74

–76

–78

–80

–72

–74

–76

–78

–80

–76

–80

–84

1.5

0.5

1.5

0.5

1.5

0.5

Water

0.2 0.2 0.3 0.02 0.04 0.06 0.08 0.10.10.4 0.6 0.8

Water

N/C+O) = 0.1

C/O = 0.54

N/(C+H) = 0.05

C/H = 0.21

N/(O+H) = 0.43

C/H = 0.39

–7

–6

–5

–4

–3

–1

–7

–6

–5

–4

–3

–1

–7

Species mol fraction

Pressure (bar)

log f(O

)

log f(O

)

log f(O

)

Water depth (m)

Species mol fractionWater depth (m)

Bulk H/(C+H+O+N) – atoms

Bulk O/(C+H+O+N) – atoms

Bulk C/(C+H+O+N) – atoms

Fig. 14.1

Species distribution in a C–H–O–N atmosphere calculated by Gibbs free energy minimization at 25

◦

C. The top

diagrams show mol fractions in the gas phase (solid curves, port axes) and oxygen fugacity (dashed curves, starboard

axes). Bottom diagrams show thickness of the liquid water column averaged over the planet’s surface, and

atmospheric pressure at the planet’s surface. Bulk content of H, O or C is varied in each plot, while keeping bulk

contents of all other components constant. Note the sharp transition between H

–NH

atmospheres and CH

–CO

atmospheres.

ways analogous to a phase transition (Chapter 7), in the sense that there are two possible

“phases”: a CO

-bearing atmosphere with very little ammonia and molecular hydrogen, or

aCO

-free atmosphere dominated by methane, ammonia and molecular hydrogen. Atmo-

spheres in which CO

,NH

and H

are all significantly abundant are not thermodynamically

stable. Over a very narrow compositional interval spanning the transition the equilibrium

species distribution consists chiefly of CH

O ±N

.Atmospheres with this composition

are unlikely to be common, however, as they require very fine-tuned conditions (Fig. 14.1).

The CO

/CH

ratio increases with the oxidation state. CO

becomes the dominant atmo-

spheric species for bulk compositions sufficiently rich in O or poor in H. Oxygen fugacity,

however, remains < ∼10

−70

bar even in atmospheres with ∼90 mol% CO

. This oxygen

fugacity is below the stability limit of hematite (e.g. Fig. 11.7 and 11.8). Thus, Fe

-rich

Archaean oceans are consistent with a CO

-dominated atmosphere (Section 11.5). Increas-

ing oxidation of a CH

–CO

atmosphere always raises atmospheric pressure, as the light

carbon species, CH

, is replaced by the much heavier CO

. The thickness of the water col-

umn, however, responds differently depending on whether oxidation is driven by hydrogen

loss, which produces CO

at the expense of liquid water and CH

, or by an increase in bulk

650 Thermodynamics of life

oxygen content, which produces liquid water and CO

at the expense of CH

. Note that the

models in Fig. 14.1 yield relatively thin water columns, less than ∼10 m, which are orders

of magnitude thinner than the terrestrial oceans. This is a function of the total volatile mass

that I arbitrarily chose as input for the calculations. The thickness of the water column can

be varied without changing atmospheric pressure nor composition by adding H and O to

the bulk composition in a proportion of 2 to 1 (Exercise 14.1).

These trends are largely unaffected by changes in bulk nitrogen content. The left panel

in Fig. 14.2 shows the effect of varying bulk N content while keeping bulk H, O and C

contents fixed at the starting values chosen above. Nitrogen-bearing species become more

abundant with increasing N content, and atmospheric pressure increases, but the relative

proportions of C–H–O species, oxygen fugacity and the depth of the liquid water column

remain constant. An ammonia-dominated atmosphere forms if the bulk composition is

sufficiently rich in both nitrogen and hydrogen. This is shown in the right panel of Fig. 14.2,

in which gas speciation is tracked as a function of variable bulk H content, for a bulk

N content ten times greater than in Fig. 14.1 (atomic ratio N/(C +O) = 1, compared to

0.2 0.4 0.6

0.5

1.5

Water

N/(C+O) =1

C/O=0.54

H/(C+O) =1.65

C/O=0.54

–6

–5

–4

–3

–2

–1

2.5

1.5

0.1 0.2

Bulk N/(C+H+O+N) – atoms Bulk H/(C+H+O+N) – atoms

0.3

Water

–7

Species mol fraction

–6

–5

–4

–3

–2

–1

–7

Species mol fraction

Pressure (bar)

–72

–74

–76

–78

–71

log f(O

)

log f(O

)

Water depth (m)

Fig. 14.2

Same as Fig. 14.1, focusing on the effects of nitrogen content. Formation of an ammonia–methane atmosphere

requires a bulk composition very rich in N and H (right panel).

651 14.1 Chemical evolution of post-nebular atmospheres

–6

–5

–4

–3

–2

–1

–7

–6

–5

–4

–3

–2

–1

–7

Species mol fraction

0 5*104 105 1.5*105

Total volatile mass – kgm

–2

Water depth (m)

Pressure (bar)

–72

–71

log f(O

)

Water

NH3

246810

Gravitational acceleration – ms

–2

Water depth (m)

0.5

Pressure (bar)

–72

–71

log f(O

)

Water

Fig. 14.3

Effects of total volatile mass and gravitational acceleration on atmospheric speciation. Atmospheric pressure varies

directly and almost linearly with both variables. Decrease in atmospheric pressure is accompanied by evaporation of

liquid water, and increase in X(H

O) in the gas phase. The effect of gravitational acceleration is strongly non-linear for

small planets, accounting at least in part for Mars’s predicament.

N/(C +O) =0.1 in the equivalent plot in Fig. 14.1). In this case too there is a rapid transition

from a strongly reducing NH

–CH

atmosphere to a N

–CO

atmosphere, in response to

hydrogen loss.

The effects of changing total volatile mass and gravitational acceleration are shown in

Fig. 14.3. Increasing both of these variables raises atmospheric pressure almost linearly

but has negligible effects on atmospheric composition, except at very low pressure, or for

O vapor content, at all pressures. The effect on X(H

O) is a consequence of saturation in

liquid water: the chemical potential, and hence partial pressure, of H

O vapor is fixed, so

its concentration must decrease as pressure increases. Decreasing pressure is accompanied

by an increase in the concentration of molecular hydrogen because the methane oxidation

reactions:

O  CO +3H

652 Thermodynamics of life

and

+2H

O  CO

+4H

both have large and positive 

V , so they are favored by low pressure. As the concentra-

tion of H

increases with decreasing pressure so does the ammonia concentration. Note in

passing that although all the calculations are performed at constant pressure, assumed to be

the pressure at the planet’s surface given by the atmospheric mass, the results in Fig. 14.3

can also be used to infer that the equilibrium species distribution does not change signifi-

cantly with elevation. The emphasis is on equilibrium because, as we saw in Chapter 12,

photoactivated processes affect non-equilibrium species distribution with elevation.

The depth of the water column varies linearly with total volatile mass, but it responds in a

non-linear fashion to changes in gravitational acceleration at constant volatile mass. In the

latter case evaporation of water (required to preserve the chemical potential of H

Ointhe

atmosphere) depletes a liquid reservoir of definite size. Of course gravitational acceleration

does not change with time once a planet has accreted. Rather, this result should be seen as

a demonstration of the difficulty that a small planet such as Mars may have had in holding

on to its oceans and lakes.

The concentrations of minor species can be calculated a posteriori, because they have a

negligible effect on the Gibbs free energy of the system and on the mass balance constraints

(equations (14.3)). For example, we can calculate the concentrations of hydrogen cyanide

and formaldehyde from the equilibria:

+CH

 HCN +3H

(i)

and:

O +CH

 CH

O +2H

(ii)

for which we have:

HCN

·X





·P

exp



−



0,(i)



·X





·P

exp



−



0,(ii)



(14.8)

Representative results are shown in Fig. 14.4 (left panel), together with CO concentrations,

which were calculated as part of the Gibbs free energy minimization procedure, but could

also have been calculated as in equations (14.8). Formaldehyde and other organic molecules

in which carbon is present in the same oxidation state (CH

O) are essential biological

building blocks (Section 14.2.1). These results suggest that their equilibrium atmospheric

concentrations are unlikely to ever have been significant, even in reducing atmospheres.

The synthesis of these molecules must have been the outcome of other processes that took

place at the inception of life on Earth.

For simplicity sulfur is not included in the model calculations, because it is likely to have

always been less abundant than C, H, O and N. Even if we ignore the absolute abundances of

sulfur species, it is important to constrain their oxidation state, partly for biological reasons.

653 14.1 Chemical evolution of post-nebular atmospheres

–10

–9

–8

–7

–6

–5

Species mol fraction

–84

–80

–76

–72

log f(O

)

HCN

0.2 0.4 0.6 0.8

–25

–24

–23

–22

–21

Species mol fraction

–84

–80

–76

–72

log f(O

)

–6

–5

–4

–3

–2

–1

X(CO

) / X (CH

)

–84

–80

–76

–72

log f(O

)

0.2 0.4 0.6 0.8

–40

–35

–30

–25

–20

–15

X(SO

) / X(H

–84

–80

–76

–72

Bulk H/(C+H+O+N)–atoms Bulk H/(C+H+O+N)–atoms

log f(O

)

Fig. 14.4

Left: atmospheric concentrations of the minor species CO, CH

O and HCN (note scale break between both diagrams).

Ammonia and molecular hydrogen shown for comparison. Right: SO

S concentration ratio, compared to CO

/CH

ratio. Sulfur is present as a reduced species even in atmospheres that have ten times more CO

than CH

We do this by means of the equilibrium:

S +O

 SO

(iii)

which yields:

·P

exp



−



0,(iii)



. (14.9)

Representative results are shown in the right panel of Fig. 14.4, where they are compared to

the CO

/CH

ratios for the same range of atmospheric compositions. An important conclu-

sion is that carbon oxidizes more readily than sulfur, so that a CO

-dominated atmosphere

can still be reducing enough to contain H

S and virtually no SO

. We will return to this

later in this chapter.

Photodissociation of CH

,NH

and H

O molecules produces free hydrogen which, as we

saw in Chapter 13, can be lost by thermal escape. This is an irreversible process, for the same

relationship between kinetic energy and gravitational binding energy (equation (13.1)) that

makes hydrogen loss possible makes it impossible for a planet to capture hydrogen atoms.

654 Thermodynamics of life

If surface conditions are such that the atmosphere contains significant H

O vapor, then

hydrogen loss leads inevitably to a dry planet with an atmosphere dominated by carbon

dioxide and nitrogen, regardless of how reduced the initial atmospheric composition might

have been (Fig. 14.1 left panel, and Fig. 14.2 right panel). In the absence of other processes a

-rich atmosphere is a terminal state, as carbon–oxygen gas species are not readily lost

by atmospheric escape processes. Protracted hydrogen loss is the most likely explanation for

the nature of the present-day atmospheres of Venus and Mars. Neither the Earth nor Titan fit

this picture, however. Titan’s atmosphere consists chiefly of nitrogen, with minor amounts

of CH

and other reduced carbon species. Thus, it must be located on the reduced side of

the speciation transition (Fig. 14.1 and 14.2). Yet according to equation (13.1) and Fig. 13.1

hydrogen loss from Titan’s atmosphere must be at least as efficient as in Venus and Mars.

Oxidation of Titan’s atmosphere is prevented by its very low surface temperature, which

keeps the partial pressure of H

O(∼ the saturation vapor pressure over ice at very low

temperature) virtually equal to 0. The example of Titan emphasizes the importance of H

vapor as the oxygen source for atmospheric oxidation. It is the only abundant molecule that

contains both H and O, so that photodissociation followed by hydrogen loss makes oxygen

available. If there is no H

O in the atmosphere, for instance because it is sequestered in

low-temperature ice, then there is simply no source of oxygen.

The Gibbs free energy minimization model also allows us to examine the effect of tem-

perature on the equilibrium species distribution. Representative results are shown in Fig.

14.5, in which I have adjusted the bulk composition so that there are ∼100 m of liquid water

in equilibrium with a CO

-rich atmosphere at 25

◦

C and ∼1.5 bar pressure. As tempera-

ture increases water must evaporate in order to preserve the equilibrium saturation vapor

pressure. This raises the concentration of H

O vapor in the atmosphere, and also atmo-

spheric mass and hence atmospheric pressure. At temperatures approaching 100

◦

C, H

becomes the dominant atmospheric component, even if water may be kept from boiling

by the high atmospheric pressure. This condition, known as a steam atmosphere, greatly

accelerates the rate of hydrogen loss because it increases both the concentration of H

in the atmosphere, and hence the rate of photodissociation (equation (12.83)), and its tem-

perature (equation (13.1)). Temperature increase also raises the equilibrium H

and NH

concentrations by several orders of magnitude.

Buildup of a steam atmosphere is thought to be a self-reinforcing process, by virtue of

the strong infrared absorption of H

O molecules (Section 13.3.6). It is possible that once

the concentration of H

O vapor exceeds certain threshold it causes a runaway temperature

increase that results in complete dessication of the planet’s surface and CO

accumulation

in the atmosphere. As long as there is liquid water CO

may be scavenged by carbonate

precipitation at a rate that is largely controlled by the rate at which silicate weathering

supplies cations such as Ca

,Mg

and Fe

in aqueous solution. This process is known

as the Urey reaction (Urey, 1952). Once liquid water disappears this scrubbing mechanism is

no longer possible, and CO

atmospheric concentration cannot decrease. Hydrogen escape

rates are likely to have always been lower in Earth than in Venus and Mars (e.g. Fig. 13.1).

Whether or not Venus ever had a steam atmosphere, it may have lost its water early on,

and with it the capability of controlling its atmospheric CO

concentration. Mars probably

underwent slower dessication and oxidation, and eventual freezing of its remaining surface

water. Hydrogen loss from Earth has been slow enough to allow much of its surface water

to persist over the age of the solar system.

If atmospheric methane concentration decreases to trace levels then the atmosphere loses

its ability to buffer oxygen fugacity. As long as there is water vapor available that can

655 14.1 Chemical evolution of post-nebular atmospheres

–7

–6

–5

–4

–3

–2

–1

Species mol fraction

0 20 40 60 80 100 120

160

170

180

Temperature - ºC

Water depth (m)

1.5

2.5

Pressure (bar)

–80

–75

–70

–65

–60

–55

log f(O

)

Water

Fig. 14.5

Response of atmospheric speciation to temperature changes. The bulk composition was adjusted so as to generate

∼180 m of liquid water in equilibrium with a CO

-dominated atmosphere at 25

◦

C. Increasing temperature raises the

mol fraction of H

O vapor in the atmosphere and atmospheric pressure. For this bulk composition H

O becomes the

dominant atmospheric component at ∼100

◦

C, giving rise to a steam atmosphere.

undergo photodissociation oxygen concentration could in principle increase freely, but this

is generally not the case. Under equilibrium conditions atmospheric oxidation is limited

by the availability of reduced species at the planet’s surface, chiefly ferrous iron, sulfides

and reduced carbon. Figure 11.8 shows that oxidation of iron to its terminal ferric state

buffers equilibrium oxygen fugacity to log(f O

) ∼−68.6 (at 25

◦

C). Sulfide oxidation can

be modeled by means of the following reaction, that produces anhydrite and hematite at the

expense of pyrite and calcite:

4FeS

+8CaCO

+15O

 8CaSO

+2Fe

+8CO

656 Thermodynamics of life

–6 –4 –20

–100

–90

–80

–70

–60

log f(O

)

log f(CO

)

pyrite + calcite

hematite + anhydrite

hematite

magnetite

O + O

+ H

T = 25º C

Fig. 14.6

Equilibrium oxygen fugacities buffered by organic matter, ferrous–ferric equilibrium, and sulﬁde–sulfate equilibrium.

and that is at equilibrium for: log(f O

) = (8/15) log(f CO

) −64.7. These two oxidation

boundaries are shown in Fig. 14.6. If a planet has internal activity and is able to resupply its

surface with reduced species such as pyrite, olivine, pyroxene and other iron silicates via

volcanism and tectonism then its equilibrium atmospheric oxygen fugacity cannot exceed

these boundaries. Venus is still active today, so that the oxygen liberated by photodissoci-

ation of H

O and hydrogen escape in the geologic past must be present in the lithosphere,

combined with iron, sulfur and other elements of variable oxidation state. There is no

O left to supply oxygen, and there is almost certainly an excess of reduced species of

internal origin. The atmosphere of Venus is a thermodynamic dead end. The case of Mars

is somewhat different. First, it still has frozen H

O on its surface, and perhaps liquid or

frozen H

O in its shallow subsurface. Second, Mars may have largely lost its capability to

transport reduced species from the planet’s interior to its surface, and the surface is already

thoroughly oxidized. The possibility therefore exists that oxygen fugacity in the Martian

atmosphere exceeds the iron and sulfur oxidation boundaries (Fig. 14.6), as attested by the

existence of perchlorates on its surface (Chapter 11).

The Earth is of course different, because its atmosphere is very far from thermodynamic

equilibrium with its surface, and in particular with the biosphere. Carbon in organic mat-

ter typically has the same oxidation state as in formaldehyde, CH

O, so that equilibrium

between organic matter and atmospheric oxygen can be modeled by the reaction:

O +O

 CO

which results in: log(f O

) = log(f CO

) −92.67. This reaction, also shown in Fig. 14.6,

is located more than 90 orders of magnitude (!) below the actual oxygen fugacity in the

terrestrial atmosphere. In a planet covered with organic matter it would not be even possible

for ferrous iron or sulfide to oxidize, if the atmosphere was in thermodynamic equilibrium

with the biosphere. Upkeep of the oxygen content of the terrestrial atmosphere is a non-

equilibrium process driven by supply of solar energy.

657 14.2 Thermodynamics of metabolic processes

14.2 Thermodynamics of metabolic processes

14.2.1 Carbon fixing and respiration of reduced organic carbon

Chemical interactions between living organisms and their substrate are of two principal

types. One of these consists of reactions that reduce oxidized carbon absorbed from the

environment, typically CO

, in which C has an oxidation state of +4. The products of these

reactions are organic molecules in which the oxidation state of C is characteristically 0.

The simplest such compound is formaldehyde: CH

O. Reactions of this type are called

carbon-fixing reactions. The reduced carbon species generated by these reactions fulfill

two distinct biological roles. Firstly, they are used by living matter to synthesize all of the

complex organic molecules needed for life – we will not discuss this role any further but

we will keep in mind that this process, which is the essence of anabolic metabolism, is the

reason why carbon fixation is essential for life as we know it. Secondly, reduced carbon

species can serve as reactants for metabolic reactions that transfer energy from the substrate

to the living organism (i.e. catabolic metabolism). These reactions are called respiration

reactions and are the other type of chemical interactions that take place between living

organisms and their substrate. Some respiration reactions use reduced carbon as a reactant.

This is most commonly organic carbon that was fixed by reactions of the first type, although

reduced carbon of inorganic origin (e.g. CH

in volcanic gases) can also be used. However,

many respiration reactions exist in which no carbon species are involved (more on this in

the next section).

Because their role is to transfer energy from the substrate to energy-carrier molecules

within living cells, a necessary condition of respiration reactions is that they must be thermo-

dynamically spontaneous. Any combination of chemical species with 

G<0 (or E > 0)

could, in principle, constitute the basis for respiration. In contrast, with one important excep-

tion that we will discuss in detail below, carbon-fixing reactions have 

G>0. They must

therefore be sustained by a constant energy flux from the environment. Perhaps the most

familiar carbon-fixing reaction is oxygenic photosynthesis, which we can model as follows:

2(g)

(l)

+hν →CH

(g)

2(g)

, (14.10)

where hν represents electromagnetic radiation in the visible part of the spectrum and

(g)

is formaldehyde gas. Actual photosynthesis produces more complex molecules

with the same oxidation state as formaldehyde, such as glucose (C

), but the

thermodynamic relations are qualitatively the same as those that can be derived from

equation (14.10). The standard state Gibbs free energy change for this reaction is

528.96 kJ per mol of formaldehyde. At equilibrium we thus have:

·f

≈10

−93

(14.11)

which, as we saw in the previous section, means that the terrestrial atmosphere (f

≈

0.2 bar) is very far from being in thermodynamic equilibrium with reduced organic matter.

Reaction (14.10) is a photochemical reaction that fixes carbon and liberates molecular

oxygen against a huge chemical potential gradient by converting an electromagnetic energy

flux to chemical bonding energy (see also Worked Example 12.3). Photosynthesis based

on other elements that can change oxidation state also exists. Two important examples are