The rate of cooling $T'$ of an object is proportional to the different of the temperature of the object and the temperature of the surroundings. ### [[../Math/Differential Equations|Differential Equation]] Form >[!seealso] AKA: The [[Heat Equation]] $\huge \frac{\partial{T}}{\partial t} \propto \frac{\partial^2{T}}{\partial x^2} $ ### Simplified Form >[!tip] This form only works if the ambient tempature $T_\alpha$ is *constant*. $\huge \begin{align} T&=(T_{0} - T_{\alpha})e^{kt} + T_{\alpha} \\ \\ T(t) &= \op{lerp}\pa{T_{\alpha}, T_{0}, e^{kt}} \end{align} $ #### Derivation Let $T(t)$ be the temperature of an object after $t$ seconds, $T_\alpha$ be the *constant* ambient temperature, an $T_0$ be the initial temperature of the object. $\huge \begin{align} \newcommand{\temp}[0]{ {\color{pink}T} } \newcommand{\dtemp}[0]{ {\color{pink}\d T} } \newcommand{\tempz}[0]{ {\color{pink}T_{0}} } \newcommand{\atemp}[0]{ {\color{skyblue}T_{\alpha}} } \newcommand{\time}[0]{ {\color{white}t} } \newcommand{\dtime}[0]{ {\color{white}\d t} } \frac{\dtemp}{\dtime} &\propto \pa{\temp - \atemp} \\ \frac{\dtemp}{\dtime} &= k\pa{\temp - \atemp} \\ \frac{\dtemp}{\temp - \atemp} &= k\dtime \\ \int \frac{\dtemp}{\temp - \atemp} &= \int k\dtime \\ \ln \left | \temp - \atemp \right| &= k\time + C \\ e^{\ln \left | \temp-\atemp \right| } &= e^{k\time + C} \\ \temp - \atemp &= e^{k\time + C} \\ \let A &= e^{C} \\ \temp - \atemp &= Ae^{k\time} \\ \temp &= Ae^{k\time} + \atemp \\ \\ \temp(0) &= \tempz \\ &=Ae^{0} + \atemp \\ \tempz - \atemp &= A \\ \\ \temp &=(\tempz - \atemp)e^{k\time} + \atemp \\ \temp(\time) &= \op{lerp}\pa{\atemp, \tempz, e^{k\time}} \end{align}$