Sir Isaac Newton investigated the cooling process of water temperature and showed that the temperature change is exponential in time and can be predicted by
Tdiff=T0e^{-kt}
Where Tdiffis the difference in temperature between the water and the air in the room,
T0 is the initial difference in the temperatures (Tt=0 – T room)
So if T is temperature of he water any time, the equation can be re-written as
T-T room=(Tt=0-Troom)e^{-kt}

Where k is constant of proportionality and t is time.

Use the equation to solve this problem:

A hard boiled egg at 98°C is put in a sink of 18°C water to cool. After 5 minutes, the eggs temperature is found to be at 38°C. Assuming that the water has not warmed up, how much longer will it take the egg to reach 20°C?

The first step is to use the data about how much the egg has cooled after 5 minutes to find the value of k, which is unknown.

(38-18) = (98-18)e^{-5k}

20 = 80e^{-5k}

20/80 = 1/4 = e^{-5k}

ln(1/4) = -5k

ln(1/4)/(-5) = k

0.277 ≅ k

Now that we have the value of k, we can find how much longer it will take the egg to cool to 20^{o}C. In this case, the egg has already cooled to 38^{o}C, so the starting or initial temp is 38^{o}C, not 98^{o}C.