How Calorimeters Turn Temperature Change into Heat

A calorimeter is useful because temperature is directly measurable, while heat transfer is inferred from the thermal response of materials with known heat capacities.

Convert temperature change into heat

For a known mass of one substance,

$$q=mc\Delta T,$$

where $m$ is mass, $c$ is specific heat capacity, and $\Delta T=T_f-T_i$.

If the calorimeter has been calibrated and is represented by an overall heat capacity,

$$q_{\text{cal}}=C_{\text{cal}}\Delta T.$$

The second term matters because the cup, vessel, thermometer, water jacket, or other apparatus can absorb heat as well as the solution or sample.

Apply energy conservation with signs

For an ideal isolated calorimetry setup,

$$q_{\text{system}}+q_{\text{surroundings}}=0.$$

If the surroundings warm, they have gained heat; the process under study has released the corresponding amount. If the surroundings cool, they have lost heat; the process has absorbed it.

Example with water

Suppose 100.0 g of water warms by 2.50 °C and $c=4.184\ \mathrm{J\,g^{-1}\, ^\circ C^{-1}}$:

$$q=(100.0)(4.184)(2.50)=1.046\times10^3\ \mathrm{J}.$$

The water gained about 1.05 kJ. If this water were the only relevant surroundings in an idealized model, the process itself would have $q\approx-1.05\ \mathrm{kJ}$.

Calibration improves the thermal model

A real calorimeter absorbs some heat and may exchange heat with the room. Calibration determines the effective calorimeter heat capacity for the actual apparatus, allowing the measured $\Delta T$ to be converted into heat without pretending that the equipment is thermally invisible.