= Temperature
{wiki}
Temperature labels <thermal equilibrium>: two internally equilibrated bodies have the same <temperature> when contact permitting only <heat> exchange produces no net <heat transfer>. The <zeroth law of thermodynamics> makes the comparison transitive. If bodies A and B separately reach <thermal equilibrium> with the same <thermometer> without appreciably disturbing either body, they agree with each other. This is what lets a <gas thermometer> compare a metal block with <water>, despite their different <internal energies> and <heat capacities>. Insulation alone makes no such comparison: it can prevent <heat transfer> between bodies at different <temperatures>. <University Physics Volume 2 by Samuel J. Ling and others/1.1 Temperature and Thermal Equilibrium>[Ling, Moebs and Sanny, University Physics Volume 2, section 1.1: temperature and the zeroth law]{ref}
The <zeroth law of thermodynamics> establishes equality comparisons, but does not choose a numerical scale. <Thermodynamic temperature> supplies an absolute scale consistent with <Carnot's theorem> and <entropy>. For a <simple compressible system> at <thermodynamic equilibrium> with fixed particle number $N$ and composition,
$$
T=\left(\frac{\partial U}{\partial S}\right)_{V,N},\qquad
\frac1T=\left(\frac{\partial S}{\partial U}\right)_{V,N}.
$$
Here $U$ is <internal energy>, $S$ is <entropy>, and $V$ is <volume (thermodynamics)>; the second equality assumes the derivative can be inverted. Temperature is a <thermodynamic conjugate variable> to <entropy> and belongs to the <intensive and extensive thermodynamic quantities> as an intensive quantity. Its <partial derivative> tells us how much <internal energy> must change per unit change of <entropy> under the stated constraints. More general systems require their other mechanical and material variables to be held fixed too. https://davidtong.org/teaching/statistical-physics/statmechhtml/S1#S1.SS2.SSS2[David Tong, Statistical Physics, section 1.2.2: temperature from the entropy derivative]{ref}
For an infinitesimal quantity $\delta Q>0$ of <heat> transferred from A to B, with no <work> or matter exchange, the equilibrium <entropy> changes give
$$
dS_{\rm total}=\delta Q\left(\frac1{T_B}-\frac1{T_A}\right).
$$
For ordinary positive <temperatures>, spontaneous <heat transfer> from $T_A>T_B$ increases total <entropy>; <thermal equilibrium> has $T_A=T_B$. This comparison uses changes in <entropy>, rather than the total <internal energy> contained in either body. A small hot object can have less <internal energy> than a large cool one. See <entropy maximization under thermal contact> for the equilibrium criterion and <negative temperature> for the special case of bounded <energy> spectra.
Express absolute <thermodynamic temperature> $T$ in <kelvin>, and a Celsius reading $t_C$ in <degree Celsius>:
$$
\frac{t_C}{{}^\circ\mathrm C}=\frac{T}{\mathrm K}-273.15,\qquad
\frac{\Delta t_C}{{}^\circ\mathrm C}=\frac{\Delta T}{\mathrm K}.
$$
The offset is exact: $300\,\mathrm K$ corresponds to $26.85\,{}^\circ\mathrm C$, and $400\,\mathrm K$ to $126.85\,{}^\circ\mathrm C$. Their absolute <temperature> ratio is $4/3$; dividing their Celsius readings gives a different number. Thus <gas thermometer> pressure ratios and <Carnot's theorem> require <thermodynamic temperature>. The modern <SI> definition of the <kelvin> fixes the <Boltzmann constant> $k_B=1.380649\times10^{-23}\,\mathrm{J\,K^{-1}}$; a reference bath realizes or transfers the scale rather than defining that constant.<The International System of Units (SI), ninth edition, SI Brochure>[section 2.3.1: kelvin and Celsius temperature]{ref}
Temperature is not universally the <mean> <kinetic energy> of a particle. In a classical, nonrelativistic <ideal gas> at <thermal equilibrium>, the <equipartition theorem> gives the mean translational <kinetic energy> relative to the gas's bulk motion:
$$
\left\langle\frac12m|\mathbf v-\mathbf u|^2\right\rangle=\frac32k_BT.
$$
Here $m$ is particle <mass>, $\mathbf v$ its <velocity>, and $\mathbf u$ the mean flow <velocity>. In this regime, different particle <masses> at the same <temperature> have equal mean translational <kinetic energies> but different characteristic speeds. A moving gas can also carry bulk <kinetic energy> without being hotter in its rest frame. Internal molecular modes and <quantum statistics> require further care; the <equipartition theorem> has classical assumptions, whereas the equilibrium <entropy> definition applies more widely. https://davidtong.org/teaching/statistical-physics/statmechhtml/S2#S2.SS2.SSS1[Tong, Statistical Physics, section 2.2.1: classical equipartition and its limits]{ref}
Back to article page
