Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 4 / 34D / Solution

Statistical temperature is defined by $1/T=(\partial S/\partial U{)}_{V,N}$, with entropy $S=k_B\log \Omega$. Maximizing total entropy when two systems exchange energy makes their derivatives equal, explaining why equal temperature characterizes thermal equilibrium and why a thermometer can measure another system.

For a dilute monatomic ideal gas, Gaussian integration of momenta gives the classical partition function $Z_N=V^N/(N!{\lambda }_T^{3N})$, with ${\lambda }_T=h/\sqrt{2\pi m{k}_{B}T}$. Differentiating its logarithm givesAt fixed volume, gas pressure is proportional to absolute temperature; at fixed pressure its volume is proportional to temperature. A low-density gas thermometer therefore supplies an absolute calibration independent of the gas species in the ideal limit. A liquid-in-glass column responds to the difference between liquid and glass thermal expansion, which need not be exactly linear. Calibration against gas temperatures and correction for expansion nonlinearities improve a simple two-point column scale; an ideal-gas law is not a law of the liquid itself.

For blackbody radiation, the electromagnetic mode density is $8\pi {\nu }^2/c^3$ per volume and frequency interval, including two polarizations. For one photon mode, summing its occupation numbers gives $Z_{\nu }={\sum }_{j\ge 0}{e}^{-jh\nu /(k_{B}T)}=(1-e^{-h\nu /(k_{B}T)}{)}^{-1}$. Differentiation of $\log Z_{\nu }$ gives mean energy $h\nu /(e^{h\nu /k_{B}T}-1)$, so the Planck distribution follows. Converting to wavelength gives spectral radianceAt its maximum, $z=hc/(\lambda k_{B}T)$ satisfies $5(1-e^{-z})=z$, whose positive root is $4.9651\dots$. Thus the Wien displacement law isThis calibrates a wavelength-based radiation thermometer and avoids identifying a subjective colour with a universal temperature. A peak per frequency interval has a different numerical location, so the spectral convention must be stated. Real objects require emissivity and atmospheric corrections; fitting a measured spectrum is more informative than choosing its visually brightest colour. Integrating the Planck distribution also gives energy density proportional to $T^4$ and radiative flux proportional to $T^4$, supplying the complementary Stefan–Boltzmann law calibration.