= Solution
\b[<Temperature> as an equilibrium relation.] Classical thermodynamics begins with the <zeroth law of thermodynamics>: <thermal equilibrium> is transitive. A <thermometer> can therefore assign a common empirical <temperature> to systems in mutual <thermal equilibrium>. Different <thermometer> properties initially give different monotone empirical scales; a resistance reading by itself does not define the absolute scale.
The <Second law of thermodynamics> provides a substance-independent thermodynamic scale. A reversible engine operating between reservoirs has a <heat> ratio depending only on their <temperatures>. One chooses absolute <temperature> so that $Q_h/Q_c=T_h/T_c$, with $Q_h,Q_c$ the positive <heat> magnitudes. Its reversible efficiency is $1-T_c/T_h$. Equivalently, reversible <heat> defines <entropy> through $dS=\delta Q_{\mathrm{rev}}/T$. For an equilibrium simple compressible system,
$$
dU=T\,dS-p\,dV+\mu\,dN,
\qquad
\boxed{\frac1T=\left(\frac{\partial S}{\partial U}\right)_{V,N}.}
$$
Classical thermodynamics uses this relation without needing a microscopic account of <entropy>. A dilute-gas <thermometer> realizes an empirical scale approaching the thermodynamic scale after corrections for nonideal behavior.
\b[Statistical meaning of the same <derivative>.] In the microcanonical description, an equilibrium <energy> shell containing $\Omega(U,V,N)$ accessible states has $S=k_B\ln\Omega$. Its statistical <temperature> is consequently
$$
\boxed{\frac1T=k_B\left(\frac{\partial\ln\Omega}{\partial U}\right)_{V,N}.}
$$
To see why this is an equilibrium <temperature>, let two weakly coupled systems exchange a small <energy> while their total <energy> stays fixed. The combined multiplicity is $\Omega_A(U_A)\Omega_B(U_{\mathrm{tot}}-U_A)$. Maximizing its <logarithm> gives
$$
\frac{\partial\ln\Omega_A}{\partial U_A}
=\frac{\partial\ln\Omega_B}{\partial U_B},
$$
which is equality of the two inverse <temperatures>. Stability requires the total <entropy> maximum, not only stationarity. The overwhelmingly dominant <macrostate> in a large system therefore realizes the <thermodynamic equilibrium> condition.
A large reservoir gives a small subsystem the canonical <probabilities> $p_i\propto e^{-\beta E_i}$, with $\beta=1/(k_BT)$. In that ensemble, $F=-k_BT\ln Z$ and $S=-k_B\sum_i p_i\ln p_i$ agree with macroscopic thermodynamic relations in their usual equilibrium regime. The equipartition identity for a classical dilute monatomic gas, $\langle p^2/(2m)\rangle=3k_BT/2$, is a useful operational connection between microscopic motion and <temperature>, but is not a universal definition: quantum modes need not obey equipartition, and a system away from equilibrium need not have one <temperature> at all.
For finite systems one must specify the <entropy> convention and <energy> resolution; surface and fluctuation corrections can distinguish definitions that agree in the ordinary <thermodynamic limit>. Bounded-energy population-inverted systems can also have a negative <entropy> <derivative> and hence a negative statistical <temperature>. That possibility lies outside an ordinary positive-temperature ideal-gas <thermometer>; it does not <mean> <temperature> is merely an arbitrary average <kinetic energy>.
\b[Obtaining the <energy>–<temperature> conversion.] The <Boltzmann constant> connects <entropy> measured in $\mathrm{J\,K^{-1}}$ with the dimensionless <logarithm> of a state count. Historically it could be determined from the <ideal gas law> $pV=Nk_BT$: measure the <molar gas constant> $R$ and independently determine the <Avogadro constant> $N_A$, then obtain $k_B=R/N_A$. The independent particle-number determination matters; using $R$ alone would not fix <energy> per particle.
An electrical method uses <Johnson-Nyquist noise>. A resistor of resistance $R_e$ at known <temperature> has, in the classical range $hf\ll k_BT$, a one-sided open-circuit voltage-noise spectrum $S_V(f)=4k_BTR_e$. Over a calibrated flat bandwidth $\Delta f$,
$$
\boxed{k_B=\frac{\langle v^2\rangle}{4TR_e\Delta f}.}
$$
A practical measurement uses an equilibrium resistor, a calibrated low-noise amplifier and a known <transfer function>, subtracts or independently characterizes amplifier <noise>, and replaces a nominal bandwidth by the actual <effective noise bandwidth>. Varying <temperature> and fitting the mean-square voltage is more informative than treating one amplified voltage sample as the thermal <variance>. Other primary methods relate <temperature> to sound speed in a dilute gas or to calibrated particle <diffusion> and <mobility>.
There is a historically relevant unit-definition qualification: by the date of this exam, the revised SI had already taken effect on 20 May 2019. The <kelvin> is now defined by fixing
$$
\boxed{k_B=1.380649\times10^{-23}\,\mathrm{J\,K^{-1}}\quad\text{exactly}.}
$$
The same experiments now realize or compare thermodynamic <temperatures> using this fixed constant, whereas historically they measured the conversion constant on an independently defined <temperature> scale. The physical distinction between equilibrium <temperature> and microscopic state counting is unchanged.https://www.bipm.org/en/measurement-units/[BIPM, SI definitions effective from 20 May 2019 and fixed <Boltzmann constant>]{ref}
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