Newton's Law of Cooling equation

[2RIP]

Problem
From Newton's Law of Cooling, we can use the differential equation

dT/dt= -k(T-T_s)

where T_s is the surrounding temperature, k is a positive constant, and T is the temperature.

Let $$\tau$$ be the time at which the initial temperature difference T₀-T_s has been reduced by half. Find the relation between k and $$\tau$$.


Work

I have found the temperature equation for the object to be

T(t) = T_s + (T₀-T_s)e^-kt

If the initial temperature difference is reduced by half, i tried

T(t) = T_s + 0.5(T₀-T_s)e^-kt

but couldn't solve it any further. Could someone please shed some light on what i should do next?

 

[jostpuur]

[
I have found the temperature equation for the object to be

T(t) = T_s + (T₀-T_s)e^-kt

This seems to be right, but here:

If the initial temperature difference is reduced by half, i tried

T(t) = T_s + 0.5(T₀-T_s)e^-kt

you made some mistake with the parameter t. If we set notation

$$\Delta T(t) = T(t) - T_s,$$

then the equation you want to solve is

$$\Delta T(\tau) = \frac{1}{2}\Delta T(0).$$

 

[2RIP]

Oh thanks for replying, i was able to get the correct answer.

k _$$\tau$$ = ln2

However, I'm still a bit unclear of why T(_$$\tau$$)-T_s and T₀-T_s canceled out during calculation. Or do T(_$$\tau$$) and T₀ equal? Also why isn't the 0.5 placed like

T($$\tau$$) = T_s + 0.5(T₀-T_s)e^-k$$\tau$$, but instead

0.5[T($$\tau$$)-T_s] = (T₀-T_s)e^-k$$\tau$$


Thanks again!

 

[jostpuur]

Oh thanks for replying, i was able to get the correct answer.

k _$$\tau$$ = ln2

The use of latex like this doesn't seem to be working very well. At least my browser makes that look like $$k^{\tau}=\textrm{ln}(2)$$, although you are most certainly meaning $$k\tau = \textrm{ln}(2)$$.

However, I'm still a bit unclear of why T(_$$\tau$$)-T_s and T₀-T_s canceled out during calculation. Or do T(_$$\tau$$) and T₀ equal? Also why isn't the 0.5 placed like

T($$\tau$$) = T_s + 0.5(T₀-T_s)e^-k$$\tau$$, but instead

0.5[T($$\tau$$)-T_s] = (T₀-T_s)e^-k$$\tau$$

It looks like you are getting the correct result, because you know what it is, and you are arriving at it "by force". Just try to chop the problem into smaller pieces, so that you can control it. We want this:

$$\Delta T(\tau) = \frac{1}{2} \Delta T(0)\quad\quad\quad\quad (1)$$

where the temperature difference at each instant is given by

$$\Delta T(t) = T(t) - T_s = (T_0-T_s) e^{-k t}.$$

So onto the left side of the equation (1) we get

$$(T_0 - T_s)e^{-k\tau}$$

and on the right side

$$\frac{1}{2} (T_0 - T_s)e^{-k\cdot 0}.$$

See how things go then?

 

[2RIP]

Oh yeah i see it now, thanks a lot jostpuur. As for the latex inputs, I think I will go look for some tutorials on this forum.