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ChemistryNat
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Homework Statement
A hot solid body is immersed into a large body of water
which is a 4 degrees C. After one hour the body's temperature is measured
at 40 degrees C, after 2 hours at 15 degrees. Assuming that Newton's law of
cooling applies and that the water body is mixed constantly and its temperature stays at 4 degrees, what was the temperature of the body initially?
Homework Equations
Newton's Law of Cooling
dT/dt=κ(T-Tm)
The Attempt at a Solution
Tm=4°C
T(1h)=40°C
T(2h)=15°C
T(0h)=?
dT/dt=k(T-Tm)
dT/dt=k(T-4)
separating variables and integrating yields
ln|T-4|= kt+c
T=ce^(kt)+4 ....eq1
T(1)=40 → 40=4+ce^k ...eq 2
T(2)=15 → 15=4+ce^2k ...eq3
So I solved for k using eq2 & eq3
36e^(2k)=11e^k
11/36=e^k
k=ln(11/36)
plugging k back into eq2
c=1296/11
eq 3 becomes :
T=(1296/11)e^(kt)+4
finding the initial temperature
T(0)=121.8°C
This answer just doesn't seem logical to me, a lot higher than I expected. I can't seem to figure out where I went wrong or if I did? Thank you! :)