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.

Simple Chemical Conversion

From the results of chemical experimentation of substance converted into another substance, it was found that the rate of change of unconverted substance is proportional to the amount of unconverted substance.

If x is the amount of unconverted substance, then

$\dfrac{dx}{dt} = -kx$

with a condition that x = xo when t = 0.

$\dfrac{dx}{dt} = -kx$

$\dfrac{dx}{x} = -k \, dt$

$\ln x = -kt + \ln C$

$\ln x = \ln e^{-kt} + \ln C$

$\ln x = \ln Ce^{-kt}$

$x = Ce^{-kt}$

From the initial condition of x = xo when t = 0
$C = x_o$

Thus,

$x = x_oe^{-kt}$

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