Previous: Part 1. Classical Thermodynamics
Temperature turned out to be the first non-mechanical physical quantity, which nevertheless is important for all theories in continuum mechanics. In his review of the history of thermodynamics (book ‘Principles of the Theory of Heat‘), Ernst Mach attributes the first formal definition of temperature to James Maxwell (1871); apparently, before Maxwell, this was considered obvious. Maxwell’s definition corresponds to the zeroth law of thermodynamics, which is usually attributed to Fowler and Guggenheim (1939).
We begin with Maxwell’s definition and then move on to the more general concept of the thermal equation of state, which is the basis of thermometry. A drawing by Sadi Carnot allows us to visualize the conceptual model and simultaneously discuss the concept of an ideal thermometer. Next, we briefly review the history of thermometry. It’s interesting to note that relatively large measurement errors in the early days played a useful role in the development of the ideal gas equation; now it serves as the basis of the temperature scale.
The statement about the existence of a thermal equation of state seems primitive: f (t, V, p) = 0, but this simple mathematical statement contains important things for experimental research. This also shows the requirements for the minimum level of mathematics necessary to understand classical thermodynamics. The final section will touch on the concept of a temperature field, which plays an important role in many theories in continuum mechanics. The role of the zeroth law in the construction of thermodynamics is also discussed.
- Maxwell on temperature
- Thermal equation of state
- From the history of thermometry
- Ideal gas equation of state and absolute temperature scale
- Mathematics of the thermal equation of state
- From temperature to temperature field
Maxwell on temperature
Below is Maxwell’s definition of temperature from his book ‘Theory of Heat‘ (1871):
‘Definition of Temperature. — The temperature of a body is its thermal state considered with reference to its power of communicating heat to other bodies.’
‘Definition of Higher and Lower Temperature. — If when two bodies are placed in thermal communication one of the bodies loses heat, and the other gains heat, that body which gives out heat is said to have a higher temperature than that which receives heat from it.’
‘Cor. If when two bodies are placed in thermal communication neither of them loses or gains heat, the two bodies are said to have equal temperatures or the same temperature. The two bodies are then said to be in thermal equilibrium.’
‘Law of Equal Temperatures. — Bodies whose temperatures are equal to that of the same body have themselves equal temperatures.’
Maxwell summarizes the practice of thermometry and provides a formal definition of temperature. It is important to note that there are two parts. The law of equal temperatures introduces the transitivity of temperature, while the definition of the highest and lowest temperatures implies the establishment of thermal equilibrium. Without this condition, the operation of a thermometer is impossible — the establishment of thermal equilibrium is a prerequisite to introduce a practical temperature scale.
However, Maxwell’s definition of temperature includes heat, which disrupt the logic of presentation, since we have not yet considered heat. Also, temperature characterizes the state of a substance, while heat characterizes a change in the state of a substance. Everything Maxwell said is true, heat transfer characterizes the establishment of thermal equilibrium, but from a logic of presentation, it is better to remove heat from the temperature definition by introducing a thermal equation of state (below, for brevity, simply the equation of state).
Thermal equation of state
Let us look again at the drawing from Sadi Carnot’s book. It is a good visual aid to examine the equation of state and thus it helps to eliminate heat from the temperature definition.
In this section, A and B are the bodies whose temperatures are to be determined, and the cylinder containing the substance under constant external pressure by the piston serves as the thermometer. It is assumed that the physical quantities of volume (or the height of the substance under the piston) and pressure are already known, and that there are methods to measure them. Then, Maxwell’s temperature definition is equivalent to the existence of the equation of state: f(t, V, p) = 0, where t is the practical temperature, the measurement scale of which remains to be chosen, V is the volume, and p is the pressure.
Connecting a thermometer to body A leads to thermal equilibrium and a change in temperature of the thermometer. According to the equation of state, this should lead to a change in volume (the height of the substance under the piston) at constant external pressure: V(t, p). The volume of the substance in the thermometer thus becomes a measure of temperature and only the choice of measurement scale is left. If moving the thermometer from A to B does not result in a volume change, then the temperature transitivity implies that the temperatures of A and B are the same.
This does not require the introduction of heat, and demonstrates the operating principle of a gas or liquid thermometer at constant pressure. The drawing also provides a conceptual model of an ideal measuring instrument. It is assumed that geometry of the housing does not change with temperature (no thermal expansion occurs in the housing), and the temperature scale is tied to the chosen external pressure.
Atmospheric pressure is typically used as the external pressure, but it is not constant over time. This change can be neglected in a liquid thermometer, but it is significant for a constant-pressure gas thermometer. Increasing measurement accuracy also requires corrections for thermal expansion of the housing of the liquid thermometer.
From the history of thermometry
Interest in quantitative temperature measurement arose in the 17th century, and the first thermometers were gas thermometers. Galileo, Otto von Guericke, Santorio, Drebbel, and others contributed to the improvement of the gas thermometer, but then work in this direction stalled. It was realized that a gas thermometer was also a barometer; in other words, its readings depended, among other things, on pressure. At that time, experiments with air pressure were just at the beginning, and work on developing a barometer as a measuring instrument was underway in parallel.
As a result, a liquid thermometer become the standard tool with alcohol or mercury as a thermometric substance. One of the first thermometers was made at the Florentine Academy (1641), followed by the work of Dalense, Halley, Huygens, Hooke, Renaldini, and others. Fahrenheit and Réaumur (1740) achieved the greatest advances to produce reliable thermometers. Various reference points and scales have been developed.
I mention the problem with the boiling point of water used as a reference point. It depends on pressure, and this was discovered fairly quickly. The solution in this case was rather simple: select the boiling point of water at a specific pressure as the reference point and introduce corrections during the calibration procedure. However, there was another serious problem: at a fixed pressure, the boiling point of water varied depending on how the boiling process was conducted. The problem was related to superheating of water, and specialized research was required to find conditions to prevent superheating.
Another problem with liquid thermometers was nonuniformity during thermal expansion. This was discovered by comparing an alcohol thermometer and a mercury thermometer. Each was calibrated using the freezing and boiling points of water, after which the scale was divided into equal intervals. The mercury and alcohol thermometers thus constructed showed different temperatures between zero and one hundred degrees Celsius. The reason for this difference was inconstancy of the coefficient of thermal expansion — it turned out to be a function of temperature. Special studies showed that mercury expands more uniformly, and mercury thermometers were adopted as a standard.
Nevertheless, the need to chose a thermometric substance raised the question of the universal temperature scale. Mercury, for example, had limitations related to its freezing and boiling points. These limits soon have been reached, and the question arose of how to measure temperatures below the freezing point and above the boiling point of mercury. Thus, in the late 18th and early 19th centuries, interest in the gas thermometer was renewed. By this time, barometers have been developed and there was hope that the equation of state for gases would allow for the introduction of the universal temperature scale.
Ideal gas and absolute temperature scale
The development of methods to measure temperature and pressure brought the opportunity to study the equation of state of gases. The invented pump allowed to control pressure, and it also became possible to control gas temperature. In this case, Sadi Carnot’s drawing above gets a different interpretation as follows. The experimenter sets external conditions (temperature and pressure) and monitors the change in volume: V(t, p). Another option is to control external conditions of temperature and volume and measure pressure: p(t, V).
Initially, only one gas, air, was known (when its composition remains constant, its behavior is the same as of an individual substance). As chemists discovered other gases, their equations of state began to be studied as well. Due to relatively large measurement errors, a universal equation of state for all gases has been assumed. In the modern form:
pV = nR(t + a)
where n is the number of moles (the ratio of mass to molecular mass), R is the universal gas constant, and a is a constant whose value depends on the choice of practical temperature scale.
However, more precise measurements by physicist Henri Victor Regnault revealed that the behavior of different gases differs from each other and from the supposed universal equation of state. At the same time, it was shown that gas behavior matches the universal equation of state for smaller pressures; as a result, the term ‘ideal gas equation of state’ was introduced in the second half of the 19th century. The advent of the kinetic theory provided an explanation — the ideal gas equation of state corresponded to a gas without interactions between molecules, which could be achieved at low concentrations (low pressures).
Despite the difference between the behavior of the ideal gas and real gases, the ideal gas equation of state was employed to construct an absolute temperature scale. The introduction of such a scale is suggested by the very form of the equation above, since at t = − a , the gas volume is zero, thus suggesting the introduction of the absolute temperature: T = t + a .
This step has been supported by classical thermodynamics – see more information in the next chapter. The ratio of the temperature functions in the practical temperature scale is associated with the maximum efficiency of a heat engine. With the absolute temperature scale, the maximum efficiency becomes equal to the ratio of absolute temperatures. What is important is that the thermodynamics absolute temperature scale coincides with the absolute temperature T in the ideal gas equation of state.
The difference in the behavior of real gases from the ideal gas equation of state requires corrections when using a real gas as the thermometric substance. The kinetic theory suggests the form of the equation of state for real simple gases, and metrologists can use a gas thermometer with real gases by introducing corrections for the non-ideal behavior. This belongs to the difficult work of metrologists, one correction after another. In any case, a gas thermometer is considered the most accurate device for measuring temperature, and the equation of state of a non-existent ideal gas serves as the basis for the accepted thermodynamic temperature scale.
Mathematics of the thermal equation of state
Let us consider the equation of state in general. This allow us to formulate the minimum mathematical requirements for understanding classical thermodynamics — the mathematical analysis of a function in two variables.
The equation of state can be expressed in three ways: V(T, p), p(T, V), or T(V, p). Formally, these are three different functions, but they are expressions of a single equation of state, since they can be transformed into each other. Solving practical problems requires knowledge of the equation of state, that is, the availability of experimental data. Conducting experiments involves determining thermal coefficients, that is, the derivatives of the equation of state.
In general, when a substance’s state changes, temperature and pressure gradients initially arise (temperature field and pressure field), but after some time, the system reaches equilibrium — the pressure and temperature values equalize and become uniform. The equation of state applies to final equilibrium states.
Let us take the differential of volume V(T, p):
Two partial derivatives show the dependence of volume on changes in temperature and pressure, respectively. There is no time in the equation above — the derivatives belongs to the uniform states of a substance.
For convenience, the derivatives are normalized, which yields thermal coefficients as follows:
| Volume expansion coefficient | |
| Isothermal compressibility | |
| Relative pressure coefficient |
Fixed values of volume and pressure (V0 and p0) are often used in the definitions above, but this is not set in the IUPAC definitions of thermal coefficients.
Overall, the three versions of the equation of state – V(T, p), p(T, V), T(V, p) – should lead to six derivatives and, therefore, six thermal coefficients. There is a relationship between them that follows from mathematics. Consider two functions, y(x, z) and x(y, z), which are obtained by transformation from each other. There is a relationship between the partial derivatives of these functions — for brevity, this will be called an inverter:
This equation allows us to find the three missing thermal coefficients from three thermal coefficients above – we simply need to take the reciprocal values.
Moreover, there are only two independent thermal coefficients due to another relation for three functions, x(y, z), y(x, z), and z(x, y), which are obtained by transformation from each other. For brevity, this will be called the permuter:
Using the permuter together with the inverter allows us to find the relationship between the three thermal coefficients above:
Let us imagine that three independent experiments were conducted to measure three thermal coefficients. The assumption of a thermal equation of state requires that the equation above hold true. If the experiments show differences beyond the measurement error, then the experimental conditions should be checked. Perhaps something went wrong somewhere. If the difference persists, then this likely means that additional variables to characterize the state of the object under study are required — three variables are not enough.
The relationships between thermal coefficients are not limited to the above. Thermal coefficients (derivatives) are actually functions — they can be considered constant only for small deviations from the measured state. The second mixed derivative of a function can be formed in two ways, and the value of the mixed derivative is independent of the order of differentiation. For the function z(x,y) as an example this means:
For clarity, I will write out the differential dz in the form where the first derivatives are written as functions f1 and f2:
Then the equality of mixed derivatives looks like as follows:
The two derivatives (two thermal coefficients) are independent in general — one cannot be derived from the other. However, there is a certain relationship between their functional dependencies due to the the mixed second derivative.
Note that among the differentials of two variables in mathematics, there are objects for which the last equality does not hold. In this case, we speak of an inexact differential, which cannot be associated with a function z(x,y). An inexact differential can be integrated, but the integral depends on the integration path, and the integral over a closed contour is not equal to zero. The last equation can be considered as a criterion for determining whether a differential is exact or inexact. In thermodynamics, such objects include heat and work — they are not functions of the state.
From temperature to temperature field
In the case of a temperature gradient, the concept of temperature is generalized by a temperature field, which assumes that each point in a continuous medium has its own temperature, with the temperature varying continuously from point to point. An example is the Fourier heat equation, in which the temperature field plays a key role. In modern physics there are limits of the temperature field application, there is a characteristic length inside which this concept becomes inapplicable.
In this section, we will consider the issue of measuring temperature in a temperature field, assuming that the thermometer size exceeds the characteristic length above. This in itself, at first glance, requires a combination of incompatibilities — the condition of establishing thermal equilibrium is imposed on the case of temperature gradients, that is, the absence of thermal equilibrium.
Let us start with the temperature at the boundary of the temperature field under consideration. Let us assume that the substance in Sadi Carnot’s drawing is simultaneously connected to two thermostats, A and B, with different temperatures applied to opposite sides of the substance. A temperature gradient is established within the substance between the two specified temperatures, but in this case, it is not a problem to imagine measurements of temperatures of the thermostats.
The next step requires a small thermometer, as liquid and gas thermometers are too big. An attempt to use them in a temperature field would lead to temperature gradients within the thermometer itself. Therefore, let us consider a thermocouple, a thermometer that operates based on the thermoelectric effect. In the ideal thermocouple model, the junction is depicted as a point, but in reality, it has well-defined dimensions that exceed the minimum characteristic length.
There are several challenges to measure temperature with a thermocouple. When the junction is connected to the substance, thermal contact occurs, which slightly alters the temperature distribution, and a slight temperature gradient exists within the finite dimensions of the junction. In this sense, the ideal measuring device is formally unattainable, but the uncertainties could be estimated and included into the measurement error.
The discussion above is based on the concept of local thermal equilibrium between the junction and the point where the temperature is measured. There is a temperature field, but in the area of the thermocouple connection, we can talk about an average temperature, which adequately characterizes the temperature in this region. This way a conceptual model of the temperature field with such limitations can be used to consider temperature measurements in the case of temperature gradients.
In any case, temperature and temperature gradients (temperature fields) are part of the theories in continuum mechanics. Probably this was one of the reasons why the term ‘zeroth law of thermodynamics’ was not introduced in the 19th century — temperature belongs not only to thermodynamics, but to continuum mechanics as well. Fowler and Guggenheim apparently forgot this fact when they introduced the term ‘zeroth law of thermodynamics’ in 1939.
In conclusion, one more important point to consider the structure of thermodynamics (see the next chapters). The establishment of thermal equilibrium is the prerequisite even to consider the first law, which examines the transition from one equilibrium state to another. The second law introduces an absolute temperature scale (T), the second law allows us to define temperature as the derivative of internal energy with respect to entropy: T = (∂U/∂S)V, and finally, the second law allows us to prove the establishment of thermal equilibrium from the Clausius inequality. Thus, the structure of classical thermodynamics seems not to be ideal, especially to mathematicians. It is important along this way not to forget about local equilibrium, the relationship between temperature and temperature fields, and the theories of continuum mechanics.
Next: Chapter 2. From Caloric Theory to Thermodynamics
References
Ernst Mach, Die Principien der Wärmelehre, 1900 (first published in 1896). There is English translation: Principles of the Theory of Heat, 1986.
R. H. Fowler, E. A. Guggenheim, Statistical Thermodynamics: A Version of Statistical Mechanics for Students of Physics and Chemistry. Cambridge, 1939.
J. C. Maxwell, Theory of Heat, 1871.
F. Rosenberger, Die Geschichte der Physik in ihren Grundzügen, mit synchronistischen Tafeln der Mathematik, der Chemie und beschreibenden Naturwissenschaften, sowie der allgemeinen Geschichte.
I have read Russian translation: History of Physics, Part two, History of Physics in Modern Times, 1933. Part three, History of Physics in the Last (19th) Century, vol. I, 1935.
P. S. Kudryavtsev, History of Physics, v. 1, From Antiquity to Mendeleev (in Russian), 1956.
Hasok Chang, Inventing Temperature: Measurement and Scientific Progress, 2004.
Additional Information
The Problem of Coordination: Temperature as a Physical Quantity: Mathematics, Physics, and Measurement. Modern Temperature Measurement. Episodes from the History of Thermometry. Helmholtz, Mach and Duhem on Measurement in Physics.
Discussion