AcademyTemperature and Heat
Academy
Calorimetry and Phase Change
Level 1 - Physics topic page in Temperature and Heat.
Principle
Calorimetry uses energy conservation to track heat exchanged between parts of an isolated system.
When a phase change occurs, energy can enter or leave a material without changing its temperature.
Notation
\(Q_i\)
heat transferred to part i of a system
\(\mathrm{J}\)
\(m\)
mass
\(\mathrm{kg}\)
\(c\)
specific heat capacity
\(\mathrm{J\,kg^{-1}\,K^{-1}}\)
\(L_f\)
latent heat of fusion
\(\mathrm{J\,kg^{-1}}\)
\(L_v\)
latent heat of vaporization
\(\mathrm{J\,kg^{-1}}\)
\(T_f\)
final equilibrium temperature
\(\mathrm{K}\)
Method
The system is split into pieces that exchange energy. Each piece may warm, cool, melt, freeze, vaporize, or condense. The heat terms are then added with signs.
Isolated exchange
\[\sum_i Q_i=0\]
Energy lost by one part is gained by another.
Temperature change
\[Q=mc(T_f-T_i)\]
Melting or vaporizing
\[Q=+mL\]
Freezing or condensing
\[Q=-mL\]
A phase change must be checked rather than assumed. First calculate whether there is enough energy to reach the phase-change temperature. Then compare the remaining energy with the latent-heat requirement.
Energy to reach phase temperature
\[Q_{\mathrm{warm/cool}}=mc(T_{\mathrm{phase}}-T_i)\]
Energy for complete phase change
\[Q_{\mathrm{phase}}=mL\]
Partial fraction
\[f=\frac{|Q_{\mathrm{available}}|}{mL}\]
If \(0<f<1\), only part of the material changes phase.
Rules
These are the compact calorimetry relations.
Isolated system
\[\sum_i Q_i=0\]
Temperature change
\[Q=mc\Delta T\]
Phase change
\[Q=\pm mL\]
Melted fraction
\[f=\frac{Q_{\mathrm{melt}}}{mL_f}\]
Examples
Question
Mix
\[0.25\,\mathrm{kg}\]
of water at \[80\,\mathrm{^\circ C}\]
with \[0.40\,\mathrm{kg}\]
of water at \[20\,\mathrm{^\circ C}\]
Find the final temperature, ignoring the container and surroundings.Answer
Both parts are water, so the same \(c\) cancels.
\[(0.25)c(T_f-80)+(0.40)c(T_f-20)=0\]
\[0.65T_f=28\]
\[T_f=43.1\,\mathrm{^\circ C}\]
Checks
- Use one final equilibrium temperature for objects that remain in thermal contact.
- During a phase change of a pure substance at fixed pressure, temperature stays constant.
- Latent heats have units of joules per kilogram, not joules per kilogram per kelvin.
- If an assumed final state gives an impossible temperature, redo the energy balance with a different phase state.