What is the differential equation for a time-constant calorimetry experiment?

[eNaught]

Hello PF. I am an undergraduate in physics doing some work in a research lab at school and I need some help with a time-constant calorimetry experiment I am working on.

I am attempting to compute the temperature as a function of time of a small sample of copper that I am heating. I am modeling the setup as if the sample, the heater, and the thermometer are all in perfect thermal equilibrium (infinite thermal conductance between all three) and some thermal resistance R to a constant temperature reservoir to which heat is lost.

I have come up with the following differential equation to describe this situation:

d/dt(Q) = C*d/dt(T) + R(T-Ts) where Q is the heat added, T is the temp of the sample/thermometer, R is the thermal resistance between the sample and the reservoir and Ts is the temp of the reservoir.

I am hoping that someone can help me determine if the above is correct and if so give me some pointers on how to solve this diff-eq. I believe this is a linear coupled system and I don't know where to start. I really appreciate the help!

 

[eNaught]

I figured it out and so I thought I would post the answer for anyone with a similar problem although I realize this is probably very trivial for most...

C*d/dt(T) = d/dt(Q) where d/dt(Q) is the difference of the power being applied from the heater an the power being lost to the reservoir.

C*d/dt(T) = V^2/Rh - k(T-Ts) where V is the voltage across the heater coil, Rh is the electrical resistance of the heater coil, k is the thermal conductance of the link from the sample to the reservoir and C is the heat capacity of the apparatus including sample, heater and thermocouple etc.