Newton's Law of Cooling

Newton's law of cooling states that the rate at which a hot object loses heat is proportional to how much hotter it currently is than its surroundings: $\dfrac{dQ}{dt}\propto-(T-T_s)$, where $T_s$ is the surrounding temperature (the negative sign shows the rate of loss itself decreases as the object cools). Combining this with $dQ=ms\,dT$ and integrating gives the exponential solution $T=T_s+b_2e^{-\frac{a}{ms}t}$, which matches the graph in Figure 8.12: cooling is fastest right at the start (largest temperature excess) and progressively slows as the object approaches room temperature, tracing a curve rather than a straight line. Example 8.8 applies this practically: knowing a hot-water sample takes 3 minutes to cool from 92°C to 84°C at a 27°C room temperature lets you predict the time it takes to cool through a diffe …