Absolute temperature

The absolute temperature or thermodynamic temperature $\theta$ is the quantity such that the ratio of absolute temperatures $\theta_{1}$ and $\theta_{2}$ of two heat reservoirs connected by a Carnot engine is

\frac{\theta_{1}}{\theta_{2}}\equiv 1- \frac{Q_{1}}{Q_{2}}=1-\eta

where $\eta$ is the thermal efficiency of the Carnot engine. It is measured in kelvins $\text{K}$ .

Since the Carnot engine is independent of the substance it is working on, so is the absolute temperature. Since $Q_{1}>0$ (the second law of thermodynamics forbids otherwise), the absolute temperature has a lower bound

\theta>0

The asymptotic bound $\theta=0$ is called the absolute zero.

From entropy#

The above definition is equivalent to the following

\boxed{\frac{1}{T}=\frac{\partial S}{\partial E}}

where $S$ is entropy and $E$ is the internal energy of the system. This is derived from Clausius' theorem and its definition of entropy in a constant-volume system (so no work $W$ ), where $dQ=dE$ and so

\frac{dQ}{T}=dS\quad\Rightarrow \quad T=\frac{dS}{dE}

Entropy could be dependent on other variables other than $E$ , so we use the partial derivative instead.

Ideal gas#

The absolute temperature coincides with the Ideal gas temperature $T=PV/Nk_{B}$ , as can be shown by using an ideal gas as the working substance for the Carnot engine. Thus, $\theta=T$ .