"Proof" of heat flow direction?

[MJCfromCT]

Hi everyone, I have a homework problem that basically says to prove that heat is conducted from a high temperature area to a low temperature area:

1. The problem statement, all variables and given/known data

Consider a one-dimensional conductor, stretching from x=0 to x=L. The two ends are maintained at T_0 and T_L. The four sides of the conductor are insulated. The temperature distribution along the conductor is steady.

q_0 represents the heat that enters through the x=0 cross section. Assume q_0 is position, so heat is conducted in the positive x-direction.

Invoke the 2nd law to prove that q_0 flows toward lower temperatures, for example, by showing that T_L cannot be greater than T_0

2. Relevant equations

2nd Law of thermodynamics


3. The attempt at a solution

My attempt is as follows:

I have the 2nd Law in the following form:

http://img338.imageshack.us/img338/9004/problem11qw8.jpg

I have come across this equation in my text (Heat Transfer, Bejan, 1993) as well:

http://img237.imageshack.us/img237/6189/problem12ks5.jpg

I wish to substitute this equation into the "q" part of the 2nd Law. From here, in order for the "dS/dt" term to be greater than or equal to zero (Entropy always increasing), the T_0 - T_L term must be greater than zero, therefore, T_0 must be greater than T_L.

Does this make sense? I'm not sure what to do about the summation term in the form of the 2nd law that I have. Do I only pick the "0" position and forget about the "L" position?

Thanks in advance for your help.

[dlgoff]

Your question got me curious, so I pulled out my old statistical physics text (berkeley physics course-volume 5 by F. Reif). Heres a couple of quotes that may help?
In an infinitesimal quasi-static process in which the system absorbs heat dQ, its entropy changes by an amount
dS=dQ/T
where T is ...called its absolute temperature.

In any process in which a thermally isolated system changes from on macrostate to another, its entropy tends to increase, i.e.,
deltaS>=0
...is significant because it specifies the direction in which nonequilibrium situations tend to proceed.

[MJCfromCT]

Thank you for the reply. I agree with the quotes you have listed, but I am unsure as to how they help. I agree that they discuss the means by which entropy increases, but I am unsure as to how the increase in entropy relates to the diffusion of heat from a high temperature area to a low temperature area.