Newton's law of cooling

Question

I learned in physics class that, according to Newton's Law of Cooling,

"The rate of fall of temperature of a body is directly proportional to the excess of temperature over the surroundings, provided that the excess is small."

i.e -$\dfrac{d\theta}{dt}$ $\propto$ $(\theta-\theta_o)$ ; where $\theta_o$ is the temperature of surroundings.

-$\dfrac{d\theta}{dt}$ =$k$$(\theta-\theta_o)$ ; $k$ is some constant.

Integrating we get $\ln(\theta-\theta_o)$=${-kt}$ (please correct me if I am wrong.)

If we need to find the time when the temperature of body is equal to the temperature of surroundings we get that $t$ tends to be infinite. That is we never get the instant when temperature of body and that of surrounding are same, then how do we say that any body is in thermal equilibrium with the surroundings? It will never happen that a body allowed to cool in surrounding will be in thermal equilibrium with surroundings.

I know something is wrong with my understanding please help.

Answer 1

Yes, this is correct. You are missing the integration-constant though. But this doesnt change your argument. Truth is this difference become exponentially small and insignificant.

But adressing this from a different perspective: we know from statistical mechanics that if you keep the energy of a system fixed (isolated body) the temperature fluctuates in a tiny amount. Same as the pressure in a fixed volume. So if the temperature difference becomes lower than the temperature fluctuations you reached equilibrium.