The calorimetric equation of mixtures

When two or more bodies at different temperatures are mixed in a thermally isolated system, after a while they all reach the same equilibrium temperature $T_e$. The principle at play is conservation of energy: the heat released by the hot bodies exactly equals that absorbed by the cold bodies, because nothing leaves the isolated system.

Calorimetric equation of mixtures

In an isolated system the sum of the exchanged heats is zero: $\ev{\sum_i m_i\,c_i\,(T_e - T_i) = 0}$ Hot bodies ($T_i > T_e$) contribute negatively (they release heat), cold bodies ($T_i < T_e$) contribute positively (they absorb heat).

Solving the equation for $T_e$ gives a weighted average of the initial temperatures, weighted by the heat capacities $m_i c_i$:

$T_e = \frac{\sum_i m_i\,c_i\,T_i}{\sum_i m_i\,c_i}$

An important physical reading follows: a body with large mass or large specific heat “weighs” more in the balance and pulls $T_e$ towards its own initial value. This is why a large mass of lukewarm water is heated very little by a small red-hot object immersed in it (see the example of the copper pot).

Watch out for kelvin

Only differences $T_e - T_i$ appear in the equation, so one can work directly in $°$C without converting. Conversion to kelvin becomes essential, on the other hand, whenever ratios of temperatures come into play (radiation, gases).

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Topics: Thermology Concepts: Specific heat and heat capacity Skills: Calorimetric equation Methods: Calorimetric equation of mixtures Objects: Calorimeter

Related exercises: Problem — Copper calorimeter · Worked exercise — Coffee and steam exchange · Problem — Four cups