Why does a heating model work?

Question

I am referring to: $T=T_0 e^{kt}$ where T=temperature,t=time and k=constant. It seems to work, I as just curios to why it works?

Answer 1

I have shown below the derivation of this model from Newton's law of heating and cooling.

If the temperature difference, ΔT, between the two bodies or between an object and the ambient is not too large, the rate of change of this temperature difference is nearly proportional to that temperature difference:

$\frac{dΔT}{dt} = - K ΔT$

$\frac{dΔT}{ΔT} = - Kdt$

$\ln(ΔT) = - Kt$

$(ΔT) = Ce^{-Kt}$

If at some time t = 0 the temperature difference is $ΔTo$,

$ΔT_{o} = Ce^{0}$

$ C = ΔT_{o}$

Thus the heating model is $(ΔT) = ΔT_{o}e^{-Kt}$

In all heat transfer modes, such as conduction, convection (except radiation) it has always been the temperature difference that governs heat transfer between tow bodies or states.

Therefore, when two bodies exchange energy thermally, the temperature of each will exponentially approach their common final equilibrium temperature.

Remark, I have taken the model to depict cooling by haveing the constant negative -K, if it were to be heating, the it is +K