specific heat capacity of degenerate electron gas

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specific heat capacity of degenerate electron gas

Postby >-) » Fri Aug 10, 2012 9:56 pm UTC

I've been trying to find the specific heat of degenerate electron gas to calculate how much heat energy is in a white dwarf, but i can't find any numbers; only equations that i don't understand. This is one of the less cryptic ones, but i still have no idea what the variables in the equations are: http://resources.metapress.com/pdf-prev ... ze=largest

Does anyone know/have any idea what the specific heat capacity of degenerate electron gas would be?

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Charlie!
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Re: specific heat capacity of degenerate electron gas

Postby Charlie! » Sat Aug 11, 2012 5:06 pm UTC

Here: http://www.cmmp.ucl.ac.uk/~ikr/3225/Section%206.pdf Equation 6.22

They do a good job explaining in the pdf. To explain the symbols on the far-right equation: Pi is pi, Kb is boltzmann's constant, T is the temperature, N is the number of electrons, and eF is the fermi energy, which can be calculated from the number of electrons and the size of the neutron star.
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Re: specific heat capacity of degenerate electron gas

Postby >-) » Sat Aug 11, 2012 7:57 pm UTC

N is the total number of electrons in the object i'm calculating for or is it electrons/cm^3?

edit: also whats the ℏ symbol stand for?

and fermi energy should be inputted in joules or eV?

in this formula for fermi energy: http://upload.wikimedia.org/wikipedia/e ... a3be1c.png

is m the mass? of a single electron?
and for N, is it the total number or per cm^3?
V is velocity?

also this shows another website with a different formula, it doesn't require the V: is it correct?
http://hyperphysics.phy-astr.gsu.edu/hb ... ermi2.html

legend
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Re: specific heat capacity of degenerate electron gas

Postby legend » Sat Aug 11, 2012 10:51 pm UTC

N is the total number of electrons in the star
V is the volume of the star
n is electron density (electrons pro volume), so we have n=N/V, that's why you don't get V in your equation
m is mass of an electron
ℏ is Planck constant

alexh123456789
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Re: specific heat capacity of degenerate electron gas

Postby alexh123456789 » Mon Aug 13, 2012 9:05 am UTC

ℏ is actually reduced planks constant (planks constant, h, divided by 2pi). As for that (de/dp)^-1 (sorry, don't know LaTeX), I've heard some of those words before, but I have no idea how you'd actually go about calculating it.

starslayer
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Re: specific heat capacity of degenerate electron gas

Postby starslayer » Mon Aug 13, 2012 3:49 pm UTC

In a white dwarf, the electrons are for all practical purposes completely degenerate, so their thermal energy does not matter when calculating the total thermal energy (they act as if T = 0). The only things that matter for this are the ions, which are not degenerate. The thermal energy of those can be approximated using the ideal gas law. Using the facts that white dwarves have basically uniform density and are almost completely isothermal, the problem becomes quite simple:
[math]\frac{U}{V} = \frac{3R\rho T}{2\mu}[/math]
where U is the thermal energy, V the volume, R is the molar gas constant, rho the density, T the temperature, and mu the mean molecular weight of the ions (for a pure carbon white dwarf, this is 6). The total thermal energy is then
[math]U = \frac{3RMT}{2\mu}[/math]All we need to know is the mass, temperature, and composition for a first approximation. For the total internal energy of a white dwarf, however, the electrons tend to matter much more than the ions.

alexh123456789
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Re: specific heat capacity of degenerate electron gas

Postby alexh123456789 » Mon Aug 13, 2012 8:52 pm UTC

starslayer wrote: mu the mean molecular weight of the ions (for a pure carbon white dwarf, this is 6).


Why is the mean molecular weight 6? Is this just the average between a carbon (12) and an electron (0) or is there something else to it (as you can tell, I don't know much about white dwarfs).

starslayer
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Re: specific heat capacity of degenerate electron gas

Postby starslayer » Mon Aug 13, 2012 10:17 pm UTC

alexh123456789 wrote:Why is the mean molecular weight 6? Is this just the average between a carbon (12) and an electron (0) or is there something else to it (as you can tell, I don't know much about white dwarfs).
Sorry, it should be 12; I was going off a faulty memory in my post. The mean molecular weight for a pure gas is basically (mass of particle in amu)/(number of particles). There's a little more nuance to it, but that's the basic idea. For neutral helium, say, the mean molecular weight is 4, while for fully ionized helium, it's 4/3. Fully ionized carbon is 12/7, by the same logic. Here, we are considering only the carbon ions, which means that mu = 12. Here's the full formula:
[math]\mu = \left( \sum_i \frac{X_i (1+Z_i)}{\mu_i} \right)^{-1}[/math]where X_i is the mass fraction of each particle species, Z_i is the number of free electrons contributed, and mu_i is the molecular weight of each species. The 1 is from the atomic nucleus/ion/atom. In the white dwarf, the number of free electrons contributed by the ions is zero, since they're all locked up and busy being degenerate.

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LaserGuy
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Re: specific heat capacity of degenerate electron gas

Postby LaserGuy » Tue Aug 14, 2012 10:41 pm UTC

alexh123456789 wrote:ℏ is actually reduced planks constant (planks constant, h, divided by 2pi). As for that (de/dp)^-1 (sorry, don't know LaTeX), I've heard some of those words before, but I have no idea how you'd actually go about calculating it.


h and ℏ are often both referred to as Planck's constant. "Reduced Planck's constant" is too much of a mouthful for the symbol that is used 99.9% of the time of the two.


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