# Thermodynamic temperature

Thermodynamic temperature is a measure, in kelvins (K) of temperature for thermodynamics, with a uniquely defined zero point at absolute zero.

A temperature of 0 K is called "absolute zero," and coincides with the minimum molecular activity (i.e., thermal energy) of matter.

Thermodynamic temperature was formerly called "absolute temperature."

In practice, the International Temperature Scale of 1990 (ITS-90) serves as the basis for high-accuracy temperature measurements in science and technology.

## Derivation of thermodynamic temperature

There are many possible scales of temperature, derived from a variety of observations of physical phenomena. The thermodynamic temperature can be shown to have special properties, and in particular can be seen to be uniquely defined (up to some constant multiplicative factor) by considering the efficiency of idealized heat engines.

Loosely stated, temperature controls the flow of heat between two systems and the universe, as we would expect any natural system, tends to progress so as to maximize entropy. Thus, we would expect there to be some relationship between temperature and entropy. In order to find this relationship let's first consider the relationship between heat, work and temperature. A heat engine is a device for converting heat into mechanical work and analysis of the Carnot heat engine provides the necessary relationships we seek. The work from a heat engine corresponds to the difference between the heat put into the system at the high temperature, qH and the heat ejected at the low temperature, qC. The efficiency is the work divided by the heat put into the system or:

[itex]

\textrm{efficiency} = \frac {w_{cy}}{q_H} = \frac{q_H-q_C}{q_H} = 1 - \frac{q_C}{q_H} [itex] (1)

where wcy is the work done per cycle. We see that the efficiency depends only on qC/qH. Because qC and qH correspond to heat transfer at the temperatures TC and TH, respectively, qC/qH should be some function of these temperatures:

[itex]

\frac{q_C}{q_H} = f(T_H,T_C) [itex] (2)

Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient. Thus, a heat engine operating between T1 and T3 must have the same efficiency as one consisting of two cycles, one between T1 and T2, and the second between T2 and T3. This can only be the case if:

[itex]

q_{13} = \frac{q_1 q_2} {q_2 q_3} [itex]

which implies:

[itex]

q_13 = f(T_1,T_3) = f(T_1,T_2)f(T_2,T_3) [itex]

Since the first function is independent of T2, this temperature must cancel on the right side, meaning f(T1,T3) is of the form g(T1)/g(T3) (i.e. f(T1,T3) = f(T1,T2)f(T2,T3) = g(T1)/g(T2)×g(T2)/g(T3) = g(T1)/g(T3)), where g is a function of a single temperature. We can now choose a temperature scale with the property that:

[itex]

\frac{q_C}{q_H} = \frac{T_C}{T_H} [itex] (3)

Substituting Equation 3 back into Equation 1 gives a relationship for the efficiency in terms of temperature:

[itex]

\textrm{efficiency} = 1 - \frac{q_C}{q_H} = 1 - \frac{T_C}{T_H} [itex] (4)

Notice that for TC=0 K the efficiency is 100% and that efficiency becomes greater than 100% below 0 K. Since an efficiency greater than 100% violates the first law of thermodynamics, this implies that 0 K is the minimum possible temperature. In fact the lowest temperature so far obtained in a macroscopic system was 20 nK, which was achieved in 1995 at NIST. Subtracting the right hand side of Equation 4 from the middle portion and rearranging gives:

[itex]

\frac {q_H}{T_H} - \frac{q_C}{T_C} = 0 [itex]

where the negative sign indicates heat ejected from the system. This relationship suggests the existence of a state function, S, defined by:

[itex]

dS = \frac {dq_\mathrm{rev}}{T} [itex] (5)

where the subscript indicates a reversible process. The change of this state function around any cycle is zero, as is necessary for any state function. This function corresponds to the entropy of the system, which we described previously. We can rearranging Equation 5 to get a new definition for temperature in terms of entropy and heat:

[itex]

T = \frac{dq_\mathrm{rev}}{dS} [itex]

For a system, where entropy S may be a function S(E) of its energy E, the thermodynamic termperature T is given by:

[itex]

\frac{1}{T} = \frac{dS}{dE} [itex]

The reciprocal of the thermodynamic termperature is the rate of increase of entropy with energy.

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