= 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 $\lambda_T=h/\sqrt{2\pi mk_BT}$. Differentiating its logarithm gives
$$
\boxed{PV=Nk_BT,\qquad U=\frac32Nk_BT}.
$$
At 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 $Z_\nu=\sum_{j\ge0}e^{-j h\nu/(k_BT)}=(1-e^{-h\nu/(k_BT)})^{-1}$. Differentiation of $\log Z_\nu$ gives mean energy $h\nu/(e^{h\nu/k_BT}-1)$, so the <Planck distribution> follows. Converting to wavelength gives spectral radiance
$$
B_\lambda(T)=\frac{2hc^2}{\lambda^5}\frac1{e^{hc/(\lambda k_BT)}-1}.
$$
At its maximum, $z=hc/(\lambda k_BT)$ satisfies $5(1-e^{-z})=z$, whose positive root is $4.9651\ldots$. Thus the <Wien displacement law> is
$$
\boxed{\lambda_{\max}T=\frac{hc}{4.9651\ldots k_B}\simeq2.898\times10^{-3}\ \mathrm{m\,K}}.
$$
This 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.
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