Douce A.P. Thermodynamics of the Earth and Planets
Подождите немного. Документ загружается.
668 Thermodynamics of life
respiration reaction that also produces formaldehyde:
nN
2
+(3n −2)H
2
+CH
4
+H
2
O →CH
2
O +2nNH
3
(14.33)
with n>6. This reaction requires a supply of the reduced species H
2
and CH
4
(e.g. volcanic
gases), liquid H
2
O, and a dense nitrogen atmosphere, which can generate a significant con-
centration of N
2
dissolved in water. Earth is a candidate planet, but this metabolic pathway
does not exist. If it arose in the primeval Earth it must have been driven to extinction by
the much more efficient methanogenesis process. Present day Titan may have the necessary
ingredients, but the liquid water requirement would restrict life based on reaction (14.33)to
deep environments, where the supply of atmospheric nitrogen may be problematic. Future
warming of Titan (as solar energy output increases) may make it more feasible, as long
as tidal and/or radioactive heating continues to support some sort of endogenous activity.
Reaction (14.33) may be a pathway to life in extra-solar worlds with less oxygen than the
early Earth, but whether the rich biology that we are familiar with could spring from these
beginnings is another question.
The possibility of life in the ice-mantled oceans of Europa, Ganymede and Enceladus is a
popular discussion topic, but the fundamental thermodynamic constraints should temper our
enthusiasm for this possibility. A source of internal heat, that undoubtedly exists in all these
three worlds, is not sufficient – what is required is a sustained free energy gradient. Leaving
aside the fanciful non-chemical possibilities discussed in the previous section, how could a
chemical potential gradient arise in an ocean covered by ice? If the ice layer is thin enough
to allow some high-energy electromagnetic radiation to get through then photodissociation
of H
2
O molecules could generate molecular oxygen, which could in turn oxidize species
such as ferrous iron or sulfide. Convection could transport these oxidized components to
the deep ocean, where a chemical potential gradient relative to reduced species would arise.
In principle some of the same respiration pathways that exist on Earth could be sustained
in this way, but the low efficiency and unpredictability of the process would make life for
putative Europan microbes rather miserable.
What I conclude, and this is opinion rather than a logically rigorous conclusion, is that
when we find extraterrestrial life we will immediately recognize it, as it will be chemically
very familiar. Yes, there are other ways to make a living, and it is possible, perhaps even
likely, that some of these other biological pathways arose in the early Earth. But there is a
good reason why we all share a common methanogenic ancestor, which is natural selection
of the most efficient metabolic pathway.
14.4 Entropy and life
Metabolism, and therefore life, is only possible if the inorganic reactants in the substrate
can be related by a chemical reaction with positive affinity. Metabolism therefore generates
entropy, just as any chemical reaction that proceeds from a state of non-equilibrium towards
equilibrium does (equation (12.77)). I am always mystified by statements to the effect that
life and biological evolution “violate” the Second Law because, in producing “ordered”
structures, they “lower the entropy of the universe”. Such statements denote a profound
ignorance of thermodynamics, as they fail to appreciate the fact that entropy is a macro-
scopic function, whereas the probabilistic foundation of entropy (the “order–disorder” idea)
669 Exercise for Chapter 14
is an exclusively microscopic concept (Chapter 4). Evolution, assembly and growth of com-
plex and organized organisms (be it a bacterium or one of my tabbies) is a macroscopic
process, and has no bearing on the state of microscopic order of the system, which is what
determines its entropy. The correct way to see any biological process is that both catabolic
metabolism (responsible for energy transformations) and anabolic metabolism (responsible
for synthesizing cellular structures from material gathered from the substrate) are chemical
reactions, and all chemical reactions generate entropy, as equation (12.77) shows.
The same is true of the origin and evolution of life, although in this case we can make an
additional argument. Say that the origin of life from inanimate matter is an ordering process,
in the same sense that growth of a crystal from a solution or melt is an ordering process – the
random distribution of atoms and simple molecules of nutrients in the “primordial soup”
becomes a more ordered arrangement of atoms in complex organic compounds, including
RNA, DNA, etc. We can use Landau theory (Section 7.6) to examine this process. We set
the order parameter φ = 0 for the inanimate precursor of life (the disordered phase), and
φ>0 for the first living organism (the ordered phase). Note that the order–disorder concept
applies to the microscopic nature of the system, as it must. Using (7.48) we write (7.49)as
follows:
G
(
φ
)
=0, (disordered phase)
G
(
φ
)
=−
1
4
g
4
φ
4
, g
4
> 0, (ordered phase)
(14.34)
and (7.50) as:
S
(
φ
)
=0, (disordered phase)
S
(
φ
)
=−
α
2
φ
2
, (ordered phase).
(14.35)
The enthalpy of ordering is given by:
H
ordering
=G
(
φ
)
ordered
+TS
(
φ
)
ordered
−G
(
φ
)
disordered
−TS
(
φ
)
disordered
=−
1
4
g
4
φ
4
−T
α
2
φ
2
< 0. (14.36)
Ordering is always an exothermic process. The system that becomes ordered (in this case,
a biological molecule) releases heat, that is absorbed by the environment. Because the
environment must be at a lower temperature in order for heat to flow, its entropy increase
is greater than the entropy decrease associated with ordering of the primordial cell. Heat
transfer generates entropy (equation (12.14)), and the “entropy of the Universe” increases.
We may not yet have a full understanding of how life on Earth originated, but we can be
certain that it did not involve intervention of some ad-hoc and imaginary entity capable of
violating the Second Law of Thermodynamics. The origin of life was a chemical reaction,
however complex, and as such it generated entropy.
Exercise for Chapter 14
14.1 Consider a planet with g =9.8ms
−2
and a water column of thickness 2 km averaged
over the surface area of the planet. The planet has an atmosphere in which the partial
670 Thermodynamics of life
pressure of N
2
is 0.8 bar. The planet’s early atmosphere was rich in CO
2
(later in
its history most of that CO
2
was scrubbed by formation of carbonates and organic
matter, but we will ignore this). Use the Maple code described in Software Box 14.1
to estimate a possible range of bulk C–H–O–N contents if the partial pressure of CO
2
in the early atmosphere was (i) 1 bar, (ii) 10 bar, (iii) 50 bar. Compare your results to
chondritic and cometary abundances, and discuss the implications for the origin and
early evolution of the terrestrial atmosphere.
1
Appendix 1 Physical constants and other useful
numbers and conversion factors
Constant Symbol Value
Gas constant R 8.3145 J K
−1
mol
−1
Avogadro’s number N 6.0221 ×10
2
mol
−1
Boltzmann constant k
B
1.3807 ×10
−23
JK
−1
Atomic mass unit u 1.6605 ×10
−27
kg
Planck constant h 6.6261 ×10
−34
Js
Speed of light in vacuum c 2.9979 ×10
8
ms
−1
Gravitation constant G 6.6726 ×10
−11
m
3
kg
−1
s
−2
Stefan–Boltzmann constant σ 5.6705 ×10
−8
Js
−1
m
−2
K
−4
Permittivity of free space /
0
8.8542 ×10
−12
C
2
N
−1
m
−2
Electron charge e 1.6022 ×10
−19
C
Electron mass m
e
9.1094 ×10
−31
kg
Million years my 3.1558 ×10
13
s
bar bar 10
5
Nm
−2
=10
5
Pa
kilobar kbar 10
8
Nm
−2
=100MPa = 0.1 GPa
671
2
Appendix 2 Derivation of thermodynamic
identities
Solving thermodynamic equations often entails simplifications and combinations of partial
derivative relationships. We often need to write equations in terms of variables that are either
easily measured or that we want to solve for, such as T , P and V , and of the three material
properties, K
T
(or β
T ,
), α and C
P
. From the material properties we extract the following
three partial derivatives, which are the ones that we wish to appear in our equations, to the
exclusion of other partial derivatives:
∂V
∂T
P
=αV (A2.1)
∂V
∂P
T
=−β
T
V =−
1
K
T
V (A2.2)
∂H
∂T
P
=C
P
. (A2.3)
There is a straightforward method that allows one to obtain any partial derivative as
a combination of these material properties and the variables P ,T and V . A “black-box”
recipe to do this was given almost a century ago by the American physicist P. W. Bridgman
(1914), who made pioneering contributions to high-pressure research. The recipe relies on
using tabulated “half identities” (see, for example, Glasstone, 1946). Bridgman’s relations
can be derived in an elegant fashion using Jacobian matrices and determinants, as discussed
by Shaw (1935) and Guggenheim (1967, p. 41). A detailed explanation of how Bridgman
obtained his equations can also be found in Tunell (1985).
Whenever possible, I find it more satisfying to derive the relations one may need from
first principles. This is the case with all thermodynamic identities used in this book. Many
of the most commonly used relations follow directly from the identity of the mixed second
derivatives of the thermodynamic potentials (Maxwell’s relations, see Section 4.9.1). Other
relations at constant T or P can be found as follows (this derivation largely follows that of
Tunell, 1985.
The idea is that thermodynamic state functions can be written as functions of the variables
P , T , V , E and S. The first task is to find expressions for the first derivatives of E and S
relative to T and P , as a function of some combination of the variables P ,V ,T and the
three material properties, α, β
T
and C
p
. The second derivatives of the Gibbs free energy
function (equations (4.135) and (4.139)) give us the first two equations:
∂S
∂T
P
=
C
P
T
(A2.4)
∂S
∂P
T
=−V α. (A2.5)
672
673 Appendix 2
Recall that (A2.5) arises from one of Maxwell’s relations (Section 4.9.1). Using (1.3.3)we
write the total differentials of E, Q and V relative to T and P :
dE =
∂E
∂T
P
dT +
∂E
∂P
T
dP (A2.6)
dQ =
∂Q
∂T
P
dT +
∂Q
∂P
T
dP = C
P
dT +
∂Q
∂P
T
dP (A2.7)
dV =
∂V
∂T
P
dT +
∂V
∂P
T
dP = αV dT −β
T
V dP . (A2.8)
We next substitute these expressions into the First Law: dE =dQ−PdV (equation (1.56))
and collect terms as follows:
∂E
∂T
P
dT +
∂E
∂P
T
dP =
(
C
P
−P αV
)
dT +
∂Q
∂P
T
+P β
T
V
dP . (A2.9)
Because dP and dT are linearly independent variables we have the following identities:
∂E
∂T
P
=C
P
−P αV (A2.10)
∂E
∂P
T
=
∂Q
∂P
T
+P β
T
V . (A2.11)
Equation (A2.10) is the third equation that we seek. To reduce (A2.11) to material properties
we use the definition of entropy, equation (4.6), to write:
∂Q
∂P
T
=T
∂S
∂P
T
=−αV T (A2.12)
so:
∂E
∂P
T
=
(
P β
T
−T α
)
V . (A2.13)
Summarizing the four equations:
∂E
∂T
P
=C
P
−P αV
∂E
∂P
T
=
(
P β
T
−T α
)
V
∂S
∂T
P
=
C
P
T
∂S
∂P
T
=−V α.
(A2.14)
As an example, suppose that we need an expression for (∂H/∂P )
T
, see Section 3.5. From
equation (4.122):
∂H
∂P
T
=
∂E
∂P
T
+P
∂V
∂P
T
+V (A2.15)
using (A2.2) and (A2.13):
∂H
∂P
T
=
(
Pβ
T
−T α
)
V −P β
T
V +V = V
(
1 −αT
)
. (A2.16)
674 Appendix 2
For a different example, suppose we need the pressure and temperature derivatives of the
Helmholtz free energy. From (4.125), (A2.13) and (A2.5) we have:
∂F
∂P
T
=
∂E
∂P
T
−T
∂S
∂P
T
=P V β
T
(A2.17)
and from (4.125), (A2.10) and (A2.4):
∂F
∂T
P
=
∂E
∂T
P
−T
∂S
∂T
P
=−P V α. (A2.18)
The equations summarized in (A2.14) work only if we seek derivatives of other thermo-
dynamic functions at constant temperature or constant pressure. If different conditions are
required, such as constant entropy, then other equations must be found, in which case it
may be convenient to use a different set of material properties, such as the adiabatic bulk
modulus or the Grüneisen parameter. Some examples are presented in the main text, see
especially Chapters 7, 8 and 9. If all else fails, then one can always use Bridgman’s formulas
or, better, Shaw’s method.
References
Acton, G., Yin, Q-Z, Verosub, K. L. et al. (2007). Micromagnetic coercitivity distributions
and interactions in chondrules with implications for paleointensities of the early solar
system. J. Geophys. Res., 112, doi:10.1029/2006JB004655.
Acuña, M. H. et al. (1999). Global distribution of crustal magnetization discovered by the
Mars Global Surveyor MAG/ER experiment. Science, 284, 790–793.
Anderson, D. L. (1989). Theory of the Earth. Blackwell, Boston, 366 pp.
Anderson, G. M. (2005). Thermodynamics of Natural Systems. Second Edition. Cambridge
University Press, Cambridge, 648 pp.
Anderson, O. L. (1995). Equations of State of Solids for Geophysics and Ceramic Science.
Oxford University Press, New York, 405 pp.
Anderson, O. L., Isaak, D. & Oda, H. (1992). High-temperature elastic constant data on
minerals relevant to geophysics. Rev. Geophys., 30, 57–90.
Asimow, P. D. (2002). Steady-state mantle–melt interacctions in one dimension: II. Thermal
interactions and irreversible terms. J. Petrol., 43, 1707–1724.
Asimow, P. D., Hirschmann, M. M., Ghiorso, M. S., O’Hara, M. J. & Stolper, E. M. (1995).
The effect of pressure induced solid–solid phase transitions on decompression melting
in the mantle. Geochim. Cosmichim. Acta, 59, 4489–4506.
Asimow, P. D. & Stolper, E. M. (1999). Steady-state mantle–melt interactions in one
dimension: I. Equilibrium transport and melt focusing. J. Petrol., 40, 475–494.
Barron, T. H. K., Berg, W. T., & Morrison, J. A. (1959). On the heat capacity of crystalline
magnesium oxide. Proc. R. Soc. London, A250, 70–83.
Baylin, M. (1994). A Survey of Thermodynamics. American Institute of Physics, New York,
657 pp.
Berg, W. T. & Morrison, J. A. (1957). The thermal properties of alkali halide crystals. I. The
heat capacity of potassium chloride, potassium bromide, potassium iodide and sodium
iodide between 2.8 and 270
◦
K. Proc. R. Soc. Lond., A242, 467–477.
Berkeley Labs Commons. https://commons.lbl.gov/display/ALSBL6/Dielectric+strength+
of+air
Berman, R. G. (1988). Internally consistent thermodynamic data for minerals in the
system Na
2
O–K
2
O–CaO–MgO–FeO–Fe
2
O
3
–Al
2
O
3
–SiO
2
–TiO
2
–H
2
O–CO
2
. J. Petrol.,
29, 445–522.
Beyer, W. H. (1987) CRC Standard Mathematical Tables. 28th Edition. CRC Press, Boca
Raton, 674 pp.
Birch, F. (1952). Elasticity and constitution of the Earth’s interior. J. Geophys. Res., 57,
227–286.
Bizzarro, M., Baker, J. A. & Haack, H. (2004). Mg isotope evidence for contemporaneous
formation of chondrules and refractory inclusions. Nature, 431, 275–578.
Bohren, C. F. & Albrecht, B. A. (1998). Atmospheric Thermodynamics. Oxford University
Press, New York, 402 pp.
675
676 References
Borg, R. J. & Dienes, G. J. (1988). An Introduction to Solid State Diffusion. Academic Press,
San Diego, 360 pp.
Bottinga, Y. (1985). On the isothermal compressibility of silicate liquids at high pressure.
Earth Pl. Sci. Letters, 74, 350–360.
Bridgman, P. W. (1914). A complete collection of thermodynamic formulas. Phys. Rev., 3,
273–281.
Brodholt, J. & Wood, B. (1993). Simulations of the structure and thermodynamic properties
of water at high pressures and temperatures. J. Geophys. Res., 98, 519–536.
Bronsted, J. N. (1922). Studies on solubility, IV. The principle of specific interaction of
ions. J. Am. Chem. Soc., 44, 877–898.
Buffett, B. A. & Bloxham, J. (2002). Energetics of numerical geodynamo models. Geophys.
J. Int., 149, 211–224.
Bullard, E. C. & Gellman, H. (1954). Homogeneous dynamos and terrestrial magnetism.
Phil. Trans. R. Soc. Lond., A 247, 213–278.
Burke, W. L. (1985). Applied Differential Geometry. Cambridge University Press,
Cambridge, 414 pp.
Burnham, C. W., Holloway, J. R. & Davis, N. F. (1969). Thermodynamic properties of water
to 1000
◦
C and 10,000 bars. Geol. Soc. Am. Spec. Paper 132,96pp.
Callen, H. B. (1985). Thermodynamics and an Introduction to Thermostatistics. Wiley, New
York, 493 pp.
Cameron, A. G. W. & Benz, W. (1991). The origin of the Moon and the single impact
hypothesis. Icarus, 92, 204–216.
Carr, M. H. (1999). Mars: surface and interior. In: Encyclopedia of the Solar System.
Eds. P. R. Weissman, L. McFadden & T. V. Johnson. Academic Press, New York,
pp. 291–308.
Carslaw, H. S. & Jaeger, J. C. (1959). Conduction of Heat in Solids. Oxford University
Press, Oxford, 510 pp.
Catling, D. C., Claire, M. W., Zahnle, K. J. et al. (2010). Atmospheric origins of perchlorate
on Mars and in the Atacama. J. Geophys. Res., 115, doi:10.1029/2009JE003425.
Cemiˇc, L. & Dachs, E. (2006). Heat capacity of ferrosilite, Fe
2
Si
2
O
6
. Phys. Chem. Minerals,
33, 457–464.
Chabrier, G., Saumon, D., Hubbard, W. B. & Lunine, J. I. (1992). The molecular–metallic
transition of hydrogen and the structure of Jupiter and Saturn. Astrophys. J., 391, 817–826.
Chamberlain, J. W. & Hunten, D. M. (1987). Theory of Planetary Atmospheres. An Intro-
duction to their Physics and Chemistry. Second Edition. Academic Press, Orlando,
481 pp.
Chambers, J. E. (2005). Planet formation. In: Meteorites, Comets and Planets. Treatise on
Geochemistry, v.1. Ed A. M. Davis. Elsevier, Amsterdam, pp. 461–475.
Chandrasekhar, S. (1958). An Introduction to the Study of Stellar Structure. Dover,
New York, 501 pp.
Chapman, S. (1930a). A theory of upper-atmospheric ozone. Mem. R. Meteorol. Soc., 3,
103–125.
Chapman, S. (1930b). On ozone and atomic oxygen in the upper atmosphere. Phil. Mag.,
10, 369–383.
Chatterjee, N. D., Kruger, R., Haller, G. & Olbricht, W. (1998). The Bayesian approach
to an internally consistent thermodynamic database: theory, database and generation of
phase diagrams. Contrib. Mineral. Petrol., 133, 149–168.
677 References
Cherniak, D. J. & Watson, E. B. (1994). A study of strontium diffusion in plagioclase using
Rutherford backscattering spectroscopy. Geochim. Cosmochim. Acta, 58, 5179–5190.
Cherniak, D. J.,Watson, E. B., Grove, M. & Harrison,T. M. (2004). Pb diffusionin monazite:
a combined RBS/SIMS study. Geochim. Cosmochim. Acta, 68, 829–840.
Chevrier, V. F. & Altheide, T. S. (2008). Low temperature aqueous ferric sulfate solutions
on the surface of Mars. Geophys. Res. Letters, 35, doi:10.1029/2008GL035489.
Chevrier, V. F., Hanley, J. & Altheide, T. S. (2009). Stability of perchlorate hydrates
and their liquid solutions at the Phoenix landing site, Mars. Geophys. Res. Letters, 36,
doi:10.1029/2009GL037497.
Choukroun, M. & Grasset, O. (2007). Thermodynamic model for water and high-pressure
ices up to 2.2 GPa and down to the metastable domain. J. Chem. Phys., 127, DOI:
10.1063/1.2768957.
Chyba, C. F., Thomas, P. J., Brookshaw, L. & Sagan, C. (1990). Cometary delivery of
organic molecules to the early Earth. Science, 249, 366–373.
Colburn, D. S. (1980). Electromagnetic heating of Io. J. Geophys. Res., 85, 7257–7261.
Cotterill, R. M. J. (1980). The physics of melting. J. Crystal Growth, 48, 582–588.
Croft, S. K., Lunine, J. I. & Kargel, J. (1988). Equation of state of ammonia–water liquid:
derivation and planetological applications. Icarus, 73, 279–293.
Cygan, R. T. & Carrigan, C. R. (1992). Time-dependent Soret transport: applications to
brine and magma. Chem. Geol., 95, 201–212.
Darken, L. S. & Gurry, R. W. (1953). Physical Chemistry of Metals. McGraw-Hill,
New York, 525 pp.
Davies, A. G. (2001). Volcanism on Io: the view from Galileo. Astronomy & Geophysics,
42, 10–15.
Davies, G. F. (1999). Dynamic Earth. Plates, Plumes and Mantle Convection. Cambridge
University Press, Cambridge, 458 pp.
de Donder, Th. (1936). L’affinite’. Gauthier-Villars, Paris, 142 pp.
de Groot, S. R. (1959). Thermodynamics of Irreversible Processes. North-Holland,
Amsterdam, 242 pp.
de Groot, S. R. & Mazur, P. (1984). Non-equilibrium Thermodynamics. Dover, New York,
510 pp.
de Pater, I. & Lissauer, J. J. (2001). Planetary Sciences. Cambridge University Press,
Cambridge, 528 pp.
Di Giulio, M. (2009). A methanogen hosted the origin of the genetic code. J. Theor. Biol.,
260, 77–82.
Donati, J.-F., Paletou, F., Bouvier, J. & Ferreira, J. (2005). Direct detection of a magnetic
field in the innermost regions of an accretion disk. Nature, 438, 466–469.
Drake, R. P. (2010). High-energy-density physics. Physics Today, 63(6), 28–33.
Duan, Z., Moller, N. & Weare, J. H. (1992). Molecular dynamics simulation of PVT
properties of geological fluids and a general equation of state of nonpolar and
weakly polar gases up to 2000 K and 20,000 bar. Geochim. Cosmichim. Acta, 56,
3839–3845.
Duan, Z., Moller, N. & Weare, J. H. (1996). A general equation of state for supercritical
fluid mixtures and molecular dynamics simulation of mixture PVTX properties. Geochim.
Cosmichim. Acta, 60, 1209–1216.
Duan, Z., Moller, N. & Weare, J. H. (1992). Accurate prediction of the thermodynamic
properties of fluids in the system H
2
O–CO
2
–CH
4
–N
2
up to 2000 K and 100 kbar from a