Newton's Law of Cooling

Newton's law of cooling states that the rate of change of the temperature of a heated (or cooled) body at any instant is proportional to the DIFFERENCE between the body's own temperature and the temperature of its surrounding medium. If θ is the temperature of the body at time t, and θ₀ is the (constant) temperature of the surrounding medium, then dθ/dt is the rate of change of temperature with respect to time, and θ − θ₀ is the temperature difference at time t. Newton's law says dθ/dt ∝ (θ − θ₀), i.e. dθ/dt = −k(θ − θ₀), where k is the constant of proportionality and the negative sign records that this temperature difference is always shrinking over time (whether the body starts hotter or colder than its surroundings).

Separating variables: dθ/(θ−θ₀) = −k·dt. Integrating both sides, and using the initial condition θ = θ₁ at t=0 to fix the constant of integration, gives θ = θ₀ + (θ₁ − θ₀)e^(−kt). This expression gives the temperature of the body at any time t, in terms of its initial temperature θ₁ and the surrounding temperature θ₀. …