Calorimetry and heat capacity

Measuring heat transfer

Calorimetry is the measurement of heat transfer. It often uses temperature changes to infer how much energy moved between objects. The central idea is energy conservation: heat lost by warmer parts equals heat gained by cooler parts, after accounting for the calorimeter and possible phase changes.

For a substance whose temperature changes without phase change,

$Q=mcDelta T$

Here $m$ is mass, $c$ is specific heat capacity, and $Delta T$ is temperature change.

Specific heat capacity

Specific heat capacity tells how much energy is required to raise one kilogram of a substance by one kelvin or one degree Celsius. Water has a large specific heat, which means it takes a lot of energy to change its temperature.

This is why oceans moderate climate and why water is useful for heating and cooling systems.

The heat capacity of an entire object is

$C=mc$

so

$Q=CDelta T$

Molar heat capacity

For gases and chemistry applications, heat capacity is often given per mole. Molar heat capacity $C_m$ satisfies

$Q=nC_mDelta T$

For gases, heat capacity depends on the process. At constant volume, no expansion work is done, so added heat changes internal energy directly. At constant pressure, some heat goes into expansion work, so more heat is required for the same temperature rise.

Thus gases have

$C_P>C_V$

For an ideal gas,

$C_P-C_V=R$

Calorimetry energy balance

In an insulated calorimeter, the total heat exchange sums to zero:

$sum Q_i=0$

For example, if hot metal is placed in cooler water, the metal loses heat and the water gains heat:

$m_m c_m(T_f-T_{m,i})+m_w c_w(T_f-T_{w,i})=0$

Solving this equation gives the final equilibrium temperature $T_f$, assuming no heat loss to the environment.

Latent heat

During a phase change, temperature can remain constant while heat is added or removed. The energy goes into changing molecular arrangement rather than changing temperature.

Latent heat is described by

$Q=mL$

where $L$ is latent heat. For melting or freezing, use latent heat of fusion. For boiling or condensing, use latent heat of vaporization.

For example, ice at $0^circ C$ can absorb heat and melt while staying at $0^circ C$ until the phase change is complete.

Heating curves

A heating curve plots temperature versus heat added. Sloped regions correspond to temperature change, modeled by $Q=mcDelta T$. Flat regions correspond to phase changes, modeled by $Q=mL$.

To solve a multi-step calorimetry problem, break the process into stages: warm the solid, melt it, warm the liquid, vaporize it, warm the gas, and so on.

Assumptions and limitations

Simple calorimetry assumes no heat is lost to the surroundings, the system reaches equilibrium, and heat capacities are constant over the temperature range. Real experiments require corrections for the calorimeter, evaporation, incomplete mixing, and heat exchange with the environment.

The big idea

Calorimetry uses energy conservation to connect heat transfer with temperature and phase changes. Specific heat describes temperature change; latent heat describes phase change. Careful energy accounting allows prediction of final temperatures, material heat capacities, and phase-change energy requirements.