MadSci Network: Astronomy

Re: Does carbon deposit on sunspots at 3825 degrees C ?

Date: Sat Jul 31 03:49:36 2010
Posted By: Vladimir Escalante-Ramirez, Faculty
Area of science: Astronomy
ID: 1279389645.As

The condensation temperature of a solid does depend on pressure. There are two ways to treat this problem: using thermodynamics, or using kinetic theory. Let us review the thermodynamics. When a substance undergoes a change of phase at constant temperature, the pressure and volume of the phases generally varies. Let us suppose that a substance condenses from gas to solid at constant temperature. The condensation takes place when gas is compressed and its pressure reaches a value called the vapor pressure. If the pressure is not increased anymore, the solid and the gas coexist in an equilibrium state. At this point molecules in the gas are getting attached to the surface of the solid while molecules of the solid leave the solid to join the vapor. The relationship between the temperature, pressure and volume of the vapor during a phase transition is given by the Clausius–Clapeyron equation. For an ideal gas this equation can be integrated to give the following equation for the pressure of the vapor:

Pvap=P0 e-T0/T

where Pvap is the vapor pressure, T is the temperature of the vapor in degrees Kelvin (K), and P0 and T0 are parameters that depend on the substance. The above formula shows that the temperature decreases with decreasing vapor pressure. To know the temperature of the gas when condensation takes place, we must equate its pressure to the vapor pressure. For an ideal gas, P=nkT, where n is the density of the gas in particles per cubic centimeter, and k=1.38×10-16erg/K is the Boltzmann constant. Therefore the condensation temperature Tc is given by substituting Pvap with the ideal gas law at that temperature:

nkTc=P0 e-T0/Tc
There is no straightforward formula to find Tc from the above equation. I suppose you know that ex is the exponential function. It is a tricky function. "Scientific" calculators have it. The easy way to find Tc is to solve the above equation for the Tc on the right hand side:
Tc=T0/ln(P0/ nkTc)
The function lnx is the natural logarithm, also available in scientific calculators. It is much gentler function than the exponential function. The way to solve this last equation is to guess an initial value for Tc on the right hand side, and calculate with a calculator a new value of Tc on the left hand side. Put this value on the right hand side, and repeat the procedure until there is no change in the value of Tc. This is called an iterative procedure to solve an equation numerically. With the unknown Tc trapped inside a logarithm function, the iterations (i.e., repetitions) converge very quickly to a value for Tc even with a rough initial guess value.

But before we do the iterations, we need values for n, P0, and T0. Vernazza, Avert and Loeser calculated solar atmosphere models in an article on the Astrophysical Journal, Vol. 45, pages 635 to 725. Their results for the pressure and density are on pages 678 to 683 of that long article. At the surface of the Sun the density is around n=1017 cm-3. Remember that the Sun is 90% hydrogen. Carbon is around 4×10-4 times the hydrogen, therefore let us take n=4×1013 cm-3. A. Evans in his book The Dusty Universe (John Wiley & Sons, 1994), p. 85, gives P0= 1.68×1014 dyn cm-2, and T0=8.888×104;K for graphite. Doing the iterations as described above, I get Tc=2930 K in two iterations. P0, and T0 will certainly be different for other forms of solid carbon like soot, but this calculation gives us an idea of the condensation temperature at the pressures of the solar surface.

The actual vapor pressure, and therefore the condensation temperature, of a solid is generally higher than the calculation above because the condensation of a solid depends on the way that molecules get stuck to form a solid. It is not the same to form a solid plane than a solid sphere. This takes us to the second way to look at the condensation of a solid from a gaseous phase. In a change of phase a solid starts forming when molecules stick together to form clusters. When the cluster gets big enough, it becomes stable against destruction by evaporation. As more molecules get stuck on the cluster, a solid grain a few microns in size starts to grow. In stellar atmospheres the growth of solid grains stops when the size of the grains is big enough to be expelled from the star by light pressure or winds.

The kinetic theory of solid formation in astrophysics shows why the condensation temperature depends on pressure. The higher the pressure, the more particles there are on the vapor to stick together and form solid grains. This situation allows a higher temperature to produce the condensation. If you want to know more on this subject, read chapter 5 of Evans' book. However the conclusion here is that solar spots are not cool enough to form solid carbon.

All of the above applies to a pure substance. The real situation in a stellar atmosphere is far more complicated because there will be a mix of elements in the gas. In the atmospheres of cool stars like red giants, carbon will combine with oxygen to form carbon monoxide (CO) among other compounds. Oxygen is more abundant than carbon in most of the universe, with an abundance about 8×10-4 times that of hydrogen. Thus all the carbon could be taken up by oxygen to form CO. However there are stars, aptly named carbon stars, where the carbon is about 10% more abundant than oxygen. In those stars there may be little atomic carbon available to form carbon grains. Atomic carbon will condense when its pressure is above the vapor pressure of solid carbon, but the pressure of carbon in the gas is given by P(C)=n(C)kT where n(C) is the density of pure carbon in the gas. If carbon atoms are "sequestered" in compounds, n(C) decreases a lot, and the condensation temperature decreases. In a typical atmosphere of a red giant star, the condensation temperature of carbon is around 1850 K. You can test this by yourself if you put a very low value for n like 106 cm-3 in the above formula and repeat the calculation.

Red stars produce solid grains in large quantities because temperatures in their atmospheres are much lower than the condensation temperatures of some solid materials like fosterite (Mg2SiO4), fayalite (Fe2SiO4), enstatite (Mg2SiO3), silicon carbide (SiC), and many others. Those solid grains are known as "interstellar dust" because they are expelled by the star and pervade all the interstellar medium. Those grains eventually form planets. In order to know the condensation temperatures of different substances in a stellar atmosphere, we need to know how abundant that substance is in order to calculate its density in particles per cubic centimeter. That is a complicated calculation that involves the chemistry of the medium. Here are some condensation temperatures at low pressure calculated by J. W. Larimer (Geochmica et Cosmochmica Acta, Vol. 31, pp. 1215 to 1238 [1967]):

SubstanceFormulaT(K) SubstanceFormulaT(K)
SpinelMgAl2O41740 FosteriteMg2SiO41420
KyaniteAl2SiO51650 GoldAu920
IronFe1620 TroiliteFeS680
WollastoniteCaSiO31580 MagnetiteFe3O4400
EnstatiteMg2SiO31470 WaterH2O210
SilicaSiO21450 MercuryHg181

Interestingly there are no carbon compounds here because most of them have very low condensation temperatures and do not form in stellar atmospheres.


Vladimir Escalante Ramírez
Center for Radio Astronomy and Astrophysics
UNAM, Morelia, Mexico

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