Derivation of Newtons 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 dTdt=−k(T−Ts)dTdt=−k(T−Ts) 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 dTdt=−k(T−Ts)dTdt=−k(T−Ts) dTT−Ts=−kdtdTT−Ts=−kdt ln(T−Ts)=−kt+lnCln⁡(T−Ts)=−kt+ln⁡C ln(T−Ts)=lne−kt+lnCln⁡(T−Ts)=ln⁡e−kt+ln⁡C ln(T−To)=lnCe−ktln⁡(T−To)=ln⁡Ce−kt T−Ts=Ce−ktT−Ts=Ce−kt when t = 0, T = To C=To−TsC=To−Ts Thus, T−Ts=(To−Ts)e−ktT−Ts=(To−Ts)e−kt T=Ts+(To−Ts)e−ktT=Ts+(To−Ts)e−kt The formula above need not be memorized, it is more useful if you understand how we arrive to the formula.