# Heat Exchange and Some Frivolous Aspects of e

Is it possible to heat a glass of \(0 ^\circ \textrm{C}\) milk to \(>50 ^\circ \textrm{C}\) using only the heat from an identical glass of \(100 ^\circ \textrm{C}\) water, thus cooling the water to \(< 50 ^\circ \textrm{C}\)? No heat is exchanged with the outside world, extra containers are available, and heat capacities per unit volume of the water and milk are assumed to be the same.

**Figure 1.**The milk is colder than the water at the beginning of the process and hotter than the water at the end. Figure courtesy of Mark Levi.

\[T_{k+1}= \frac{n}{n+1} T_k, \ \ T_0 = 100 ^\circ \textrm{C}, \tag1 \]

since the heat of \(n\) units of water spreads equally among the \(n+1\) units of liquid.

After \(n\) steps,** ^{1}** with all of the milk in the third glass, the water therefore cools to

\[\frac{100}{\bigl(1+ \frac{1}{n} \bigl) ^n } \approx\frac{100}{e} \approx 36.8 ^\circ \textrm{C};\]

coincidentally, this is the human body temperature. The milk’s temperature is thus \(\approx 63 ^\circ \textrm{C} \), considerably above \(50 ^\circ \textrm{C}\). This is actually the perfect temperature for cooked salmon. To summarize,

\[T_{\rm body}+T_{\rm salmon}\approx 100 ^\circ \textrm{C}\]

and

\[e\approx \frac{100 ^\circ \textrm{C}}{T_{\rm body}}.\]

Even more surprising than the reversal of the order of temperatures is the fact that a near-perfect temperature swap is possible in principle. Biological evolution “invented” the mechanism of such a swap, which may be described in another article.

** ^{1}** We assume a small ladle, hence a large \(n\).

**References**

[1] Levi, M. (2012). *Why Cats Land on Their Feet: And 76 other Physical Paradoxes and Puzzles*. Princeton, NJ: Princeton University Press.