The concept and reality of absolute temperature is amazing. It tells us something fundamental about the universe, including physical limits as to what is possible. The existence of absolute temperature is intimately connected with the existence of entropy as a thermodynamic state function. It also hints at the underlying quantum nature of reality.
Aside: Unfortunately, the Wikipedia page on this topic is mediocre and garbled. For example, it continues the myth that temperature is related to kinetic energy.
The zeroth law of thermodynamics allows the definition of empirical temperature. It is an equilibrium state variable that indicates whether a thermodynamic system will remain in the same state upon being brought into thermal contact with another system. Thermometers are systems with a single state variable.
Absolute temperature is a specific temperature scale that is central to thermodynamics and statistical mechanics.
There are several equivalent definitions of absolute temperature. They start at different points. Except for the first one, the others show that the existence of absolute temperature is intimately connected to the second law and to entropy being an extensive quantity.
This is nicely discussed by Zemansky in chapter 8 of his text Heat and Thermodynamics, Fifth Edition (1968). [This was the text for my second year undergrad thermo course at ANU in 1980. At the time, I did not fully appreciate how profound some of it is. I just enjoyed all the multivariable calculus.]
1. Ideal gas thermometers.
Consider a fixed mass of ideal gas whose volume is fixed. An ideal gas is defined as any gas at a temperature and pressure much larger than the critical temperature and pressure for the gas-liquid transition. Suppose the system is cooled and heated, and the pressure is measured as a function of the temperature measured by a separate thermometer calibrated by the Celsius scale. The pressure versus temperature curve is a straight line. If this line is extrapolated to zero pressure, this occurs at -273.15 degrees Celsius. The straight line has different slopes for different gases, but they all intercept the x-axis at the same point. Alternatively, one can take the pressure as fixed and measure the volume of the gas versus temperature. Extrapolation to zero volume also occurs at -273.15 degrees.
This suggests that something special is happening at -273.15 degrees Celsius. One can define a special temperature scale where this temperature is zero. Historically, this was the beginning of the concept of absolute temperature.
However, we should be cautious about this approach. This is just an extrapolation and does not allow for the fact that ideal gases are rather special or that some very different physics might kick in below the critical temperature of helium.
2. The efficiency of Carnot cycles.
This follows Zemansky (page 208). Consider a Carnot cycle abcda, where b to c and d to a are isothermal processes, between the same two reversible adiabatic surfaces, and involve heat transfers Q and Q_3, respectively. The absolute temperature scale T is defined by
T/T_3 = Q/Q_3
with T_3 = 273.16, when the process d to a occurs at the triple point of water.
3. Integrating factor for heat
Heat is not a state property. It depends on processes. The first law says Delta Q = Delta U + P Delta V. If we consider a quasi-static process and integrate the heat transfer along the path taken (in state space), the result may depend on the path taken. On the other hand, if one integrates dQ/T, one finds that the result is independent of the path. This can then be used to define a new state variable, the entropy.
The brief discussion above misses some subtle and profound features that only became clear in the 1960s following the work of Pippard, Turner, Landsberg, and Sears, which was inspired by an axiomatic approach to thermodynamics developed by Caratheodory.
Zemansky states
It is an extraordinary circumstance that not only does an integrating factor exist for the dQ of any system, but this integrating factor is a function of temperature only and is the same function for all systems! This universal character enables us to define an absolute temperature.
4. Applying the second law to a composite system
This treatment follows Schroeder, Thermal Physics (Section 3.1)
Schroeder defines entropy in terms of a multiplicity of states. However, I prefer to define entropy as the state function which tells us whether or not two states are accessible from one another by an adiabatic process. There are multiple possible versions of this empirical entropy state function, but let's choose one that is extensive, i.e., scales with the mass and volume of the system.
Consider an adiabatically isolated system containing an internal partition through which the conduction of heat can occur. Denote the two parts of the system by A and B. The entropy of each part can be written as a function U of its internal energy.
The total entropy of the system can be written
S = S_A (U_A) + S_B (U_B)
If the system is in thermal equilibrium, by the second law, the entropy of the whole system must be a minimum as a function of U_A and U_B.
Now, dU_A = - dU_B as the composite system is adiabatically isolated. Hence, we have.
The left-hand (right-hand) side of the equation only depends on the properties of system A (B). Thus, it is an intensive state variable which determines whether the system will be in equilibrium with another system. Hence, by the zeroth law, it defines a temperature scale.
T is the absolute temperature.
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