State Newton's law of cooling and derive its mathematical equation.

(N/A) According to Newton's law of cooling,the rate of loss of heat,$-\frac{dQ}{dt}$,of a body is directly proportional to the temperature difference $\Delta T = (T - T_s)$ between the body and its surroundings,provided the difference is small.
Mathematically,$-\frac{dQ}{dt} = k(T - T_s) \dots (1)$
where $k$ is a positive constant depending on the surface area and nature of the body.
If a body of mass $m$ and specific heat capacity $s$ is at temperature $T$,the heat lost $dQ$ for a small temperature change $dT$ is $dQ = ms dT$.
Therefore,the rate of loss of heat is $\frac{dQ}{dt} = ms \frac{dT}{dt} \dots (2)$.
Equating $(1)$ and $(2)$:
$-ms \frac{dT}{dt} = k(T - T_s)$
$\frac{dT}{T - T_s} = -\frac{k}{ms} dt$
Let $K = \frac{k}{ms}$,then $\frac{dT}{T - T_s} = -K dt$.
Integrating both sides:
$\int \frac{dT}{T - T_s} = -\int K dt$
$\ln(T - T_s) = -Kt + C$
$T - T_s = e^{-Kt + C} = C' e^{-Kt}$
$T(t) = T_s + C' e^{-Kt}$
This equation describes how the temperature of the body decays exponentially towards the ambient temperature $T_s$.