Calorimetry

Energy conservation principle applied to isolated systems with heat transfer between bodies at different temperatures.

Conceptual context

Calorimetry is the field of thermal physics that studies and quantifies the exchange of heat between bodies. It is the most direct application of the energy conservation principle to thermal systems: if a set of bodies at different temperatures is isolated from the surroundings, the energy that some release is exactly the energy that others absorb.

This leaf comes after the concepts of $Q_i$ (heat as energy in transit), $c_i$ (specific heat capacity as a material property), and $\Delta T_i$ (temperature change), and before the study of phase changes, where calorimetry is extended to include latent heats. Calorimetry simultaneously brings together all the magnitudes from earlier leaves and integrates them into a single principle.

The central question is deceptively simple: to what temperature will two bodies at different temperatures arrive when placed in thermal contact inside an isolated container? The answer is not the arithmetic mean of the temperatures, but a weighted average where the weight of each body is its total heat capacity $m_i$ · $c_i$.

🟢 Essential level

When hot water is mixed with cold water in an insulated cup, both reach an intermediate temperature. If there is much more hot water than cold, the final temperature is nearly as high as the hot water. If there is much more cold water, the final temperature is close to the cold water. The more abundant water "wins" because it has more capacity to store energy.

The calorimetry principle expresses this idea precisely: in an isolated system, total energy is conserved. The heat released by the bodies that cool down is exactly equal to the heat absorbed by those that warm up. This is written by saying that the sum of all exchanged heats is zero.

If a hot body releases heat, its heat has a negative sign (it loses energy). If a cold body absorbs heat, its heat has a positive sign (it gains energy). The zero-sum condition ensures that energy is neither created nor destroyed.

🔵 Formal level

Basic formulas

The calorimetry principle is expressed as:

Calorimetry principle (isolated system)

Where each $Q_i$ is the heat of the i-th body of the system, calculated using the sensible heat relation with its specific heat capacity $c_i$, its mass $m_i$, and its temperature change $\Delta T_i$.

Sensible heat of body i

For a two-body system without phase change, the equilibrium temperature is obtained by solving for $T_{eq}$ from the zero-sum condition:

Equilibrium temperature for two bodies

Real student questions

Why isn't it the arithmetic mean of the temperatures? The arithmetic mean would be correct only if both bodies had the same total heat capacity. In general, a body with greater $m_i$ · $c_i$ needs more heat to change its temperature by one kelvin and therefore "pulls" the equilibrium more strongly toward its initial temperature. The weighted average reflects this imbalance.

What happens if one of the bodies changes phase during the process? The process is no longer pure sensible heat. One must calculate how far the cooling or heating proceeds before the transition (using sensible $Q_i$), then apply the latent heat of the transition, and then continue with sensible heat in the new phase. If the available energy is not sufficient to complete the transition, the system reaches equilibrium at the transition temperature with part of the material in each phase.

How is the water equivalent of the calorimeter measured experimentally? A calibration experiment is performed by mixing hot water of known mass and temperature with cold water of known mass and temperature inside the calorimeter. The difference between the measured $T_{eq}$ and the calculated one without the calorimeter is due to the heat absorbed by the calorimeter. Equating that difference to the sensible heat of an equivalent water mass determines that equivalent.

Final synthesis

Calorimetry applies energy conservation to isolated thermal systems: the algebraic sum of all exchanged $Q_i$ is zero. The equilibrium temperature $T_{eq}$ is the weighted average of initial temperatures with weights equal to the total heat capacities $m_i$ · $c_i$ of each body. The principle is valid without phase changes; with them, the process is divided into stages. The zero-sum condition is always the definitive self-check criterion.