Newton's Law of Cooling

Newton's Law of Cooling states that the temperature of a body changes at a rate proportional to the difference in temperature between its own temperature and the temperature of its surroundings.
 

We can therefore write

$\dfrac{dT}{dt} = -k(T - T_s)$

where,
T = temperature of the body at any time, t
Ts = temperature of the surroundings (also called ambient temperature)
To = initial temperature of the body
k = constant of proportionality
 

$\dfrac{dT}{dt} = -k(T - T_s)$

$\dfrac{dT}{T - T_s} = -k \, dt$

$\ln (T - T_s) = -kt + \ln C$

$\ln (T - T_s) = \ln e^{-kt} + \ln C$

$\ln (T - T_o) = \ln Ce^{-kt}$

$T - T_s = Ce^{-kt}$
 

when t = 0, T = To
$C = T_o - T_s$
 

Thus,
$T - T_s = (T_o - T_s)e^{-kt}$

$T = T_s + (T_o - T_s)e^{-kt}$

The formula above need not be memorized, it is more useful if you understand how we arrive to the formula.
 

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