Temperature and heat
Temperature and heat are related but fundamentally different. Temperature describes the thermal state of a system; heat is energy crossing a system boundary because of a temperature difference. Confusing the two causes many errors in calorimetry and phase-change problems.
Temperature as an energy scale
For an ideal monatomic gas, average translational kinetic energy is proportional to absolute temperature. In condensed matter the connection is more complicated because energy is also stored in vibrations, rotations and interactions.
Two objects at the same temperature are in thermal equilibrium even if they contain very different total internal energies. A bathtub of warm water can contain far more thermal energy than a tiny red-hot spark.
Sensible heat and heat capacity
q = mcΔT
m is mass and c is specific heat capacity. The equation applies over temperature ranges where c is approximately constant and no phase change occurs.
Heat capacity is extensive for an entire object; specific heat capacity is per unit mass; molar heat capacity is per mole.
Latent heat and phase changes
At a first-order phase transition under fixed pressure, energy can be absorbed or released without changing temperature:
q = nΔHtransition
During melting, energy disrupts the solid structure. During vaporization, it separates molecules against intermolecular attractions. The kinetic-temperature picture alone is therefore incomplete.
Calorimetry as energy conservation
In an insulated calorimeter, the sum of heat exchanges is approximately zero:
qhot + qcold + qcalorimeter = 0.
The final equilibrium temperature must lie between the initial hot and cold temperatures if no reaction or phase change supplies additional energy.
Real experiments need corrections for heat loss, container heat capacity and thermometer response.
Kelvin, Celsius and temperature differences
A temperature difference of 1 K equals a difference of 1 °C, so either unit can be used for ΔT in q = mcΔT. Absolute-temperature equations such as gas laws and entropy formulas require kelvin.
0 K is the thermodynamic zero point. It is not “no energy whatsoever”: quantum systems can retain zero-point energy.
Heat-transfer mechanisms
Conduction transfers energy through microscopic interactions within matter; convection transports energy with moving fluid; thermal radiation carries energy electromagnetically and requires no material medium.
Many real systems combine all three. A cooling metal object in air conducts internally, convects heat to surrounding air and radiates infrared energy.
From microscopic motion to thermal engineering
Heat capacity reveals internal degrees of freedom
A monatomic ideal gas stores thermal energy mainly in translation, while molecules can also store energy in rotation and vibration when those modes are thermally accessible. This is why molar heat capacities differ among substances and can vary with temperature.
Solids also show temperature-dependent heat capacities: at low temperature many vibrational modes are frozen out, while at higher temperature more modes participate.
Thermal expansion is not “molecules getting bigger”
Heating usually increases average atomic separation because interatomic potential-energy wells are asymmetric. The atoms themselves do not swell. The effect is small in many solids but crucial in bridges, rails, precision instruments and bimetallic strips.
Rate of cooling is a transport problem
Thermodynamics predicts the final equilibrium temperature; heat transfer predicts how quickly it is approached. Newton’s law of cooling is often a useful approximation when convective heat loss is roughly proportional to the temperature difference from the surroundings.
Geometry, airflow, emissivity and thermal conductivity can change the cooling time dramatically without changing the equilibrium state.
Worked example: mixing water
Mix 100 g of water at 80 °C with 100 g at 20 °C in an ideal insulated container. Because both portions have the same mass and heat capacity, energy conservation gives a final temperature of 50 °C.
If the hot portion were copper instead of water, the final temperature would be much closer to the initial temperature of the water because copper has a much smaller specific heat capacity. Equal masses do not imply equal thermal capacities.
Energy during boiling
At 1 atm, liquid water can remain near 100 °C while energy is supplied because that energy goes into vaporization. Temperature begins rising again only after the phase change is complete or pressure changes.
Exercises
Heat capacity
How much heat raises 200 g of water by 10 K if c = 4.18 J g⁻¹ K⁻¹?
Solution
q = 200 × 4.18 × 10 = 8.36 kJ.
Phase plateau
Why can temperature stay constant while ice melts?
Solution
Added energy changes the phase structure rather than increasing average thermal kinetic energy.
Kelvin
Can °C be used directly in PV = nRT?
Solution
No. The gas law requires an absolute temperature in kelvin.