5.3 - Thermodynamics

To warm $0.40\,\mathrm{kg}$ of ice with $c=2100\,\mathrm{J\,kg^{-1}\,K^{-1}}$ through $15\,\mathrm{K}$ requires $12.6\,\mathrm{kJ}$. Melting that mass then requires a separate $L\Delta m$ calculation. Add energies for successive stages; do not combine their equations into one substitution.

Place an NTC thermistor and fixed resistor in series across a stable d.c. supply. Measure the potential-divider output across the chosen component with a voltmeter, while the thermistor and a calibrated thermometer or temperature probe sit close together in a stirred water bath.

An NTC thermistor's resistance falls as temperature rises. If output is measured across the thermistor, $V_{out}=V_sR_T/(R_T+R)$ falls; measured across the fixed resistor, it rises. A comparator or relay can switch heating when the calibrated threshold is crossed.

For latent heat of vaporisation, use an immersion heater to keep a liquid boiling steadily. Measure heater voltage and current, and use a balance to determine the mass lost during a timed interval after steady boiling has begun. A lid or insulation reduces energy transfer to the surroundings without sealing the vessel.

A substance's internal energy is the total randomly distributed kinetic energy and intermolecular potential energy of its molecules. Random molecular motion contributes kinetic energy; relative molecular positions and interactions contribute potential energy.

On a cooling curve, a sloping section shows falling temperature and therefore falling average molecular kinetic energy. A constant-temperature phase-change section can still show decreasing internal energy because molecular potential energy decreases.

For an ideal gas, the average translational kinetic energy of its molecules is directly proportional to absolute temperature: $\langle E_k\rangle\propto T$. Equal absolute temperatures therefore mean equal average translational kinetic energy, even for gases with different molecular masses.

The kelvin scale starts at absolute zero: $0\,\mathrm{K}=-273.15\,^{\circ}\mathrm{C}$. Convert with $T/\mathrm{K}=\theta/^{\circ}\mathrm{C}+273.15$. A rise of $1\,\mathrm{K}$ is the same size as a rise of $1\,^{\circ}\mathrm{C}$, but the scale origins differ.

When temperature rises, molecules have greater mean-square speed and momentum, so wall collisions transfer momentum more frequently or more strongly. This microscopic change explains increases in gas pressure or volume under the relevant macroscopic constraint.

Use pressure $p$ in pascals, volume $V$ in cubic metres, absolute temperature $T$ in kelvin, number of molecules $N$, and Boltzmann constant $k=1.38\times10^{-23}\,\mathrm{J\,K^{-1}}$. The equation models an ideal gas in equilibrium.

For $N=5.0\times10^{22}$ molecules at $300\,\mathrm{K}$ occupying $1.8\times10^{-3}\,\mathrm{m^3}$, $p=NkT/V=1.15\times10^5\,\mathrm{Pa}$. Convert litres to cubic metres and use the number of molecules, not the number of moles.

Trap a fixed mass of gas in a sealed syringe or Boyle's-law apparatus connected to an absolute pressure sensor. Change the volume in measured steps, secure the piston, and wait for the pressure to stabilise before recording each pair.

Repeat readings across the safe range. Either calculate $pV$ for several pairs and look for a constant value, or plot $p$ against $1/V$: a straight line through the origin supports $p\propto1/V$ at fixed temperature.

Kinetic theory gives $pV=\tfrac13Nm\langle c^2\rangle$, where $m$ is one molecule's mass and $\langle c^2\rangle$ is the mean square speed. The ideal gas equation gives $pV=NkT$. Equating them and cancelling $N$ produces the required molecular-energy relation.

The left side is the average translational kinetic energy per molecule. It can also be written as $3pV/(2N)$. For $p=1.15\times10^5\,\mathrm{Pa}$, $V=1.77\times10^{-3}\,\mathrm{m^3}$ and $N=5.15\times10^{22}$, the average is $5.93\times10^{-21}\,\mathrm{J}$.