# Newton's Law of Cooling Problem

## Homework Statement

A small metal bar, whose initial temperature was 20 degrees C is dropped into a large container of boiling water. How long will it take the bar to reach 90 degrees C if it is known that its temperature increases 2 degrees in 1 second. How long will it take the bar to reach 98 degrees C?

(the back of the book gives 82.1 seconds and 145.7 seconds)

## Homework Equations

$$\frac{dT}{dt}=k(T-Tm)$$

where $$T_m$$ is the temperature of the surroundings.

## The Attempt at a Solution

$$\int{\frac{1}{T-T_m}\frac{dT}{dt}dt=\int kdt$$

$$T(t)=Ce^{kt}+T_m$$

I'm given the points $$T(0)=20$$ and $$T(1)=T(0)+2=22$$

The problem is that this give two equations and three unknowns:

$$C+T_m=20$$

$$Ce^k+T_m=22$$

I'm given that the water is boiling, but that only gives a minimum temperature of 100 degrees celsius.

Quiablo
Actually if the water is boiling you can be sure what the temperature is, as it will be a function of the atmospheric pressure only. Assuming the experiment is performed at the sea level, the temperature of the boiling water shall be 100 C.

Ok, you're right! I actually haven't taken chem yet. This is a differential equations application problem. Are you getting this from $$PV=nRT$$ ?I guess if you have $$T=\frac{PV}{nR}$$ I guess I can see how T is a function of pressure assuming constant volume.
I should have just tried using 100 C, because it solves the system and gives the correct function $$T(t)=-80e^{tln(.975)}+100$$ and t=82.1 gives 90 C.