Thermal conductivity and Debye temperature

[Flucky]

Hi all

Homework Statement

The Debye temperature of argon is 92 K and that of silicon is 345 K. Rank the following in order of thermal conductivity (largest value first):
(i) A 1 cm3 cube of silicon at 6 K
(ii) A 512 mm3 cube of silicon at 2 K
(iii) A 1 mm3 cube of argon at 4 K
(iv) A 512 mm3 cube of argon at 2 K

You may assume that the argon and silicon are pure (i.e. there are no defects or impurities).What additional assumptions have you made?

Homework Equations

[1] κ = $\frac{1}{3}$vl$\frac{C_{v}}{V_{m}}$

where v is speed of sound [3], l is mean free path, C$_{v}$ is molar heat capacity and V$_{m}$ is molar volume.

[2] C$_{v}$ = $\frac{12π^{4}}{\hbar}$Nk$_{b}$($\frac{T}{θ_{D}}$)$^{3}$

where N is number of atoms, T is temperature and θ$_{D}$ is Debye temperature.

[3] v = $\frac{θ_{D}k_{b}}{\hbar}$$\sqrt[3]{\frac{V}{6π^{2}N}}$

The Attempt at a Solution

The thing that is throwing me is N, number of atoms. There is no density or mass given in the question so I'm not sure what to do. One idea was to assume that N is proportional to V but I don't know how to incorporate that into the equations. Maybe there is something I can do with V/N together instead of treating them separately.

Also the molar volume V$_{m}$ surely I need a mass or density in order to find out how many moles there are for each part of the question?

I'm also not sure what to do with the mean free path, l.

Any pointers in the right direction would be really appreciated.

 

[Nate810]

Additional assumptions: 1. The argon and silicon are pure and free from defects or impurities. 2. The molar volume of the argon and silicon is constant. 3. The mean free path of the argon and silicon is constant.