Are specific heat and thermal conductivity related? 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?
 A: 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:


*

*Yes, Aluminium will be able to store more thermal energy than Iron (http://www.engineeringtoolbox.com/specific-heat-metals-d_152.html) per mass.

*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)
A: TLDR - heat capacity comes from how phonons behave in the harmonic approximation, while thermal conductivity comes from how anharmonic phonons are.
In more detail, thermal conductivity comes primarily from two channels:
(1) Conduction electrons carrying heat
(2) Anharmonicity of the crystal lattice (i.e. phonons scattering from one another because the crystal lattice isn't a perfect quadratic potential).
On the other hand, near ambient temperature, the main contributer to the heat capacity is the harmonic behavior of phonons (i.e. their frequency, which is their energy cost to excite).
Generally, there is no rule relating the harmonic to anharmonic terms of the lattice potential/phonons, so the heat capacity and thermal conductivity typically have very different behavior and knowing one does not usually tell you much about the other.
A: 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).
A: 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 $10$.
A: 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 $
A: I've been pondering the same question and as a chemist with a knowledge of thermodynamics I may be able to help. Heat capacity is mathematically defined thermodynamic value i.e. $C_p=(dH/dT)p=T(dS/dT)p$ 
The units when intensive are Joules/Kelvin so you can think of it as the energy required to change the temperature. 
I know less about thermal conductivity which seems to a more engineer centered term, but one equation I found is $k=(n<v>\lambda C_v)/3Na$
 where $C_v$ is heat capacity at constant volume. Thermal conductivity relates to how heat is transfered within a material and as you can see, the definition includes heat capacity as a term. 
A: There is no connection between Conductivity and specific heat. In your case, I would go with aluminum vessels. But lets have a deeper look:
The problem starts when you say "better retention". The "better" depends always on the application. If you want a better vessel, you probably want a material to transfer as efficiently as possible the heat generated from the source to the meals. In this case, you want a material with very good conductivity since most of the heat generated will pass through the material. In this case, the thermal capacity of the vessel doesn't matter. 
In the case that you want to have a constant temperature in your pot so that to not loose its temperature when you drop inside your meat, then you need a material with a good combination of specific heat and conductivity. The value that you are mostly interested in this case is called "Thermal Diffusivity" which in essence is like a thermal inertia. Check Wikipedia's article for more information.
Finally, regarding the mass, the more mass you have the more easily the vessel will retain its temperature.
A: No. Specific heat and thermal conductivity are not related.
As Aluminum has higher of both of those values compared to Iron this is how it will work in real world.
If you have a pan made out of Al and one from Iron (steel or CI) at 20 degrees. If you heat it with a source at 150 degrees then Al will absorb more heat than Iron after a given time thats because of Higher thermal conductivity. However, the temperature of the Iron will be higher even though less heat is absorbed thats due to higher specific heat.
Hope that explains it 
