﻿ Thermodynamic temperature

# Thermodynamic temperature

Absolute temperature, thermodynamic temperature, is a temperature scale, relating to the physically justified absolute zero. It is a fundamental concept of thermodynamics and physical chemistry. As part of the International System of Units, it is measured in Kelvin, in the U.S., the Rankine scale is used.

Since absolute zero is the lowest possible temperature that can only be achieved theoretically (see the third law of thermodynamics ), the Kelvin scale a ratio scale dar. Some other temperature scales, however, refer to an arbitrarily defined zero point, as the Celsius scale whose origin was originally the freezing point of water, which is after the Kelvin scale at 273.15 K.

## Thermodynamic definition

Thermodynamic temperature of a physical system in thermal equilibrium is formally defined in terms of the efficiency of heat engines. The following two requirements define the thermodynamic temperature.

• First, we define the ratio of temperatures as follows: One considers a reversible and periodically operating heat engine that a reservoir A removes a ( infinitesimally small ) amount of heat in a period, converts part of it into mechanical work, and the rest as waste heat to a reservoir B outputs. The two reservoirs A and B are thereby are each in different thermal equilibrium states. ( In this case, both negative and positive signs for permitted, depending on whether A is colder or warmer than b. ) The ratio of the temperatures of A and B, is then defined as follows:
• By choosing a temperature reference point, the thermodynamic temperature is then fully defined. For example, one chooses in the SI unit system as the reference point: The triple point of water is defined as having the thermodynamic temperature of 273.16 K (Kelvin).

The temperature behind this definition empirical observation is that two heat engines that work in the competition for the best efficiency between two given thermal baths each constant temperature, have a similar efficiency. The more both parties strive to minimize energy losses of their machine, the lower your still possible increases in efficiency and the lower the differences between the competitors. The remarkable thing is that this also applies if the Functioning of the competing machines as diverse as steam turbine, Stirling engine and Peltier element. Thus, this definition has the advantage of universality. At any given temperature, a physical process using high efficiency there are selected at low temperatures such as magnetic effects, see Magnetic cooling.

### Derivation from the general gas law

Also from the behavior of ideal gases can be concluded that the absolute temperature.

The absolute temperature can be represented as a limit:

Where p is the pressure, v is the molar volume, R is the gas constant. At the threshold pressure to zero the gas particles show no interaction with each other, which is also referred to as an ideal gas.

### Logical consistency of the temperature Definition

The logical consistency of the temperature definition is the result of the second law of thermodynamics. It is namely:

• Two reversible and periodically operating heat engine between the same reservoirs A and B have exactly the same efficiency. Otherwise you could namely the heat engine with the lower efficiency run "backwards" as a heat pump, the machine with the higher efficiency but forward, and in a way that in the balance sheet of the reservoir B is fed the same amount of heat as is taken. They would then have a total of a cyclically operating machine, which removes only the reservoir A heat gains from mechanical work, however, reservoir B leaves unchanged. That would be a perpetual motion machine of the second kind, which does not exist according to the second law of thermodynamics.
• Consider three reservoirs A, B and C, each for itself in thermal equilibrium. The above definition then provides three temperature ratios, and. Thus, the temperature definition is consistent, the following consistency condition must apply:

### Statistical definition of entropy and

The statistical definition of temperature by Boltzmann is the absolute temperature in a correlation with the entropy that (ie the phase space volume) indicates a logarithmic measure of the number of an isolated system accessible microstates for a given macrostate:

Wherein the proportionality factor denotes the Boltzmann constant. Is the absolute temperature is then the inverse of the partial derivative of the entropy with respect to the internal energy:

For all reversible interactions in which only heat is exchanged, the following applies:

Resulting

And the formulation by Clausius follows:

The icon characterizes an incomplete differential.

## The temperature in statistical mechanics

Closely related to this notion of thermodynamic temperature is the temperature in statistical mechanics: A system of statistical mechanics in thermal equilibrium at temperature T is described by a probability density. Where H denotes the energy function, ie in the classical physics, the Hamiltonian function, in quantum physics the Hamiltonian. Further refers to the Boltzmann constant. The normalization constant Z is called the partition function. The term is called Boltzmann factor.

## Apparently negative values

However, finding negative absolute temperatures than pure computational aids quite application. So you can, for example, quite simply describe the state of a population inversion with this tool. This is only possible because this is not to state in thermodynamic equilibrium. Ideas about it were pursued already in the 1950s by Edward Mills Purcell and Robert Pound and Norman Ramsey.

## Logarithmic scale

Rudolf Plank suggests in "Handbook of Refrigeration ", alternatively, a logarithmic temperature scale is present in which no " lowest possible " temperature occurs. The zero point corresponds to the melting point of ice. Among the sub-zero temperatures to minus extend infinitely.

" [ ... ] If now the magnetic field away suddenly, enters the thermomagnetic cooling effect. In this way obtained with alum at a temperature of 0.05K. In 1935 one is even already penetrated to 0,005 K. [ ... ] In order to assess the progress made correctly, one would have to apply the logarithmic temperature scale, as has been suggested by Lord Kelvin actually. After this would come the same meaning a reduction of 100 K to 10 K, as [ ... ] of 1 K to 0.1 K. "

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