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Thermodynamic temperature

An intensive state variable T equalized at thermal equilibrium; formally defined by 1/T = (∂S/∂U)_{V,N} and linking heat to entropy.

Thermodynamic temperature

Thermodynamic temperature $T$ is an intensive variable that characterizes thermal equilibrium. The zeroth law of thermodynamics asserts that thermal equilibrium defines an equivalence relation among systems, allowing each equivalence class to be labeled by a single parameter called temperature.

A precise equilibrium definition is obtained from entropy. If equilibrium states admit a fundamental relation $S(U,V,N)$ , then temperature is defined by

\frac{1}{T} = \left(\frac{\partial S}{\partial U}\right)_{V,N},

using the notion of partial derivative . Equivalently, in the energy representation $U(S,V,N)$ ,

T = \left(\frac{\partial U}{\partial S}\right)_{V,N}.

In any reversible process , temperature is also the integrating factor that converts the inexact heat differential into an exact entropy differential:

\delta Q_{\mathrm{rev}} = T\, dS.

Physical interpretation

Temperature controls the direction of spontaneous heat flow: when two systems are placed in thermal contact through a diathermal wall , energy tends to flow as heat from the system at higher $T$ to the one at lower $T$ until thermal equilibrium is reached.

A thermal reservoir is an idealized large system whose temperature remains approximately constant while exchanging finite heat, making it a practical reference for defining and measuring $T$ .

Key relations and examples

Absolute temperature scale: temperature defined from the zeroth/second-law structure is the absolute temperature scale (Kelvin scale), independent of any particular material thermometer model.
Ideal-gas equation of state (example):
$P V = N k_B T,$
connecting pressure $P$ , volume $V$ , particle number $N$ , and Boltzmann’s constant $k_B$ through an equation of state .
Inverse temperature: statistical mechanics often uses β (inverse temperature) , defined by $\beta = 1/(k_B T)$ , especially in contexts aligned with the canonical ensemble convention .
Heat capacities: temperature dependence of energy is captured by response functions such as heat capacity at constant volume and heat capacity at constant pressure , which quantify how much energy (or enthalpy) must be supplied as heat to raise $T$ under specified constraints.
Sign and consistency: interpreting formulas like $dU = \delta Q - \delta W$ requires a fixed work sign convention , since temperature links directly to heat through $\delta Q_{\mathrm{rev}} = T\, dS$ .