About this laboratory

What is being simulated

Every instrument here is a mathematical model. What makes the exercise worthwhile is that the models are the real ones: the same equations of state, partition functions and transport relations you meet in the lectures, evaluated numerically to close to machine precision, and then degraded by realistic instrument noise, finite resolution and apparatus-specific systematic errors.

The readings you take are computed on the server. The accepted values used by the marking scheme are never sent to your browser, and the apparatus is rebuilt from your bench key on every request, so the answers cannot be read out of the page.

The physical models

Real gases

The virial equation truncated after the second coefficient. B(T) is obtained by numerically integrating the Lennard-Jones (12-6) potential, with the potential parameters that Hirschfelder, Curtiss and Bird fitted to experimental second-virial data. The same B(T) drives the gas thermometer, the free expansion and the Joule-Thomson apparatus, so those three experiments are mutually consistent.

Heat capacities of gases

The rigid-rotor, harmonic-oscillator ideal-gas model evaluated from spectroscopic constants. The rotational partition function is summed explicitly where the rotational quantum is not small, which matters for hydrogen; nuclear-spin statistics are included for the frozen $\frac{ortho}{para}$ mixture found in a cylinder.

Heat capacities of solids

A Debye lattice with a weakly temperature-dependent Debye temperature, a Sommerfeld electronic term, and the exact thermodynamic difference between $C_p$ and $C_V$, with the expansivity tied to the heat capacity by the Gruneisen relation so that it vanishes correctly as T tends to zero.

Saturation curves

Water uses the Wagner and Pruss equation adopted by IAPWS; the organic liquids use their published Antoine constants. The vapour is treated as ideal, which is stated in the manual and is itself one of the things the experiment measures.

Radiation

Planck's law integrated over the instrument's triangular slit function, so the finite spectral bandwidth broadens the measured spectrum and shifts the apparent peak, exactly as it does in a real monochromator.

Kinetic theory

A direct-simulation Monte Carlo relaxation of a hard-sphere gas. Each collision conserves momentum and kinetic energy identically, and the post-collision relative velocity is isotropic in the centre-of-mass frame, which is exact for hard spheres.

What the models cannot do

A simulation is only as honest as its stated limits. The truncated virial equation fails at high density, so the apparatus refuses to operate there rather than returning a wrong answer. A single Debye temperature cannot reproduce both the room-temperature heat capacity and the standard entropy of a real metal, and the resulting few per cent discrepancy is left in place and discussed rather than tuned away. The manual gives a full account of every such limitation.

Constants and data

The seven defining constants of the SI are exact. Measured constants are the CODATA 2022 recommended values. Material data are the accepted figures from the CRC Handbook and from Kittel; molecular constants are standard spectroscopic tabulations. Every number is cited at the point of use in the source code.

Syllabus coverage

The eleven experiments between them cover 30 distinct topics from the PHYS 4703 course outline:

• Application of Gibb's function to phase changes and Clausius-Clapeyron equation
• Application of Maxwell's relations
• Boltzmann statistics and distribution
• Bose-Einstein distribution
• Carnot's cycle and theorem
• Concept of entropy
• Consequences of the third law
• Density of quantum states
• Einstein's and Debye's theory of heat capacity of a solid
• Entropy and latent heat
• Entropy of an ideal gas
• Entropy, probability and disorder - the Boltzmann relation
• Heat capacity at constant volume and constant pressure
• Heat engines and the second law of thermodynamics
• Heat, work and internal energy
• Helmholtz and Gibb's free energy
• Isothermal and adiabatic changes for ideal gas
• Maxwell's relations
• Maxwell-Boltzmann distribution
• Measuring entropy and entropy changes
• Microscopic and Simon formulation of the third law
• Quantum states and energy levels
• Special cases of the first law of thermodynamics
• Statement of the first law of thermodynamics
• Statement of the zeroth law
• Statements of the third law of thermodynamics
• The Fermi-Dirac distribution
• The Kelvin temperature scale
• The principle of increasing entropy
• Thermodynamic potentials: internal energy, enthalpy, Helmholtz function and Gibb's function

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