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Are there any logical relationship between specific heat capacity and thermal conductivity ?

I was wondering about this when I was reading an article on whether to choose cast iron or aluminium vessels for kitchen.

Aluminium has more thermal conductivity and specific heat than iron ( source ).

This must mean more energy is required to raise an unit of aluminium than iron yet aluminium conducts heat better than cast iron.

Does it mean that aluminium also retains heat better ?

How does mass of the vessel affect the heat retention?

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Your conclusions are wrong, contarary to reality. In practice, You have to compare the masses of the two pans and specific heat. Conduction is irrelevant for Your question. To Your headline: No, at least not a simple one. –  Georg Oct 27 '11 at 9:49
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@Georg how is conduction irrelevant to a question about the relationship between conduction and specific heat? –  Nathaniel Apr 6 '13 at 17:31
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4 Answers

This is in address to your last two questions:

  • Which retains heat better?
  • How does mass effect heat retention?

You introduce two material properties (mass, specific heat) that seemingly affect the heat retention but do not give you the whole picture in of themselves. However, we can combine them to give us a useful measure of heat retention; this is also known by many other names (thermal mass, volumetric heat capacity, thermal capacitance).

Thermal Mass $C_{th} = m C_p $

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Imagine a substance in the size and form of an ice cube. If you could keep shooting it with a photon of say energy $1$ and you shot $10$ of these photons and noticed that the substance had gained a temperature difference say from $25$ to $26^\circ\mathrm{C}$, then its specific heat capacity would be $10$. (Specific heat capacity is more like a measure of the external energy given to produce the temperature change.) And it might even give off this temperature as fast as it got it.

Now for thermal conductivity (this guy is more like a range thing). If you could place a finger on one side of this substance and start your photon shooting on the other side, you may notice that even if the photon-receiving side has done the $25$ to $26^\circ\mathrm{C}$ climb, the side your finger is on might not have. (What you're doing now is obtaining the thermal conductivity of that substance.) $20$ photons might get the climb or not. Going on to $30$, $40$, ......

So basically to obtain this climb for the same cube of aluminium or iron, it might take $10$.

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Hi ogbans, and welcome to physics stackexchange! Your posts are more likely to be well-received (hence upvoted), and are also more likely to be useful to others, if you use proper spelling and grammar. (It's the internet, I know, but take a look at the other posts.) Also, if you want to format math (e.g. degree symbols) you can use LaTeX-style markup - click the edit button on your post to see how I did this (there are other ways too). –  Chris White Apr 6 '13 at 20:51
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There is not really a general answer to your question because both, the specific heat capacity and the thermal conductivity are not due to a single process in the material.

Both are in general terms a "sum" over the individual components in the material that can store thermal energy or transport thermal energy.

For metals at room temperature the most important terms of these sums are the electrons and phonons (vibrations of the lattice). Both can store and transport thermal energy. Their exact values, temperature dependence, etc. is highly material specific.

The specific heat part that is due to the electrons is mainly governed by electrons within a certain energy range (the Fermi energy). Exactly the same electrons transport heat in the material. So more electrons in that range means both, more specific heat and a higher thermal conductivity.

This get complicated if you look at a real material. A little bit of impurities or defects will influence the thermal conductivity quite a bit but the specific heat will not be influenced significantly.

In your concrete case:

  1. Yes, Aluminium will be able to store more thermal energy than Iron (http://www.engineeringtoolbox.com/specific-heat-metals-d_152.html) per mass.
  2. The mass will linearly increase the heat capacity, more mass, higher heat capacity.

(I did not use your term retention, because it is not really defined, but thermal conductivity and heat capacity are easy to understand)

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At room temperatures in metals electrons contribute almost nothing to the specific heat. Typical Fermi temperature is about 40K Kelvin. Means - thermal excitations are almost negligible. –  valdo Oct 31 '11 at 15:14
    
The phonon contribution will surely dominate, the electrons will only contribute a few percent at room temperature. They are a nice example though that thermal conductivity and specific heat are connected. The connection cannot be expressed by a single law like Wiedemann-Franz because the specific heat is not really influenced by scattering processes. –  Alexander Oct 31 '11 at 17:35
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For metals there is a connection between the thermal conductivity and electric conductivity (Wiedemann–Franz law).

However specific heat is not directly related. This is because electric and thermal conductivity are due to the electrons, however the specific heat is mostly due to the ion vibrations (phonons).

Despite "classical" intuition electrons contribute almost nothing for specific heat in metals. Electrons in a typical metal behave close to an ideal fermion gas, in a very deep quantum range (typical Fermi temperature is about 40K Kelvins).

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1st this is named Wiedemann-Franz-Lorenz and 2nd this is a rule, not a law! and 3rd this isn't an answer. –  Georg Oct 27 '11 at 21:28
    
@Georg: (1) Wiedemann–Franz law is a common name for this phenomena, see here: en.wikipedia.org/wiki/Wiedemann%E2%80%93Franz_law –  valdo Oct 28 '11 at 6:11
    
@Georg: (2) Why isn't it "an answer"? If specific heat is contributed mostly by phonons, whereas thermal/electric conductivity is due to the electrons? –  valdo Oct 28 '11 at 6:13
    
Wiki and american "science" isn't reliable for such questions. It is called a rule, because it is not really universal and precise. And not an answer: reread the question! –  Georg Oct 28 '11 at 10:54
    
@Georg: IMHO in physics, unlike math, there are no "universal and precise" laws. Virtually every law has its scope of application. The very fundamental laws eventually get reformulated upon new discoveries. –  valdo Oct 28 '11 at 16:41
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