Previous: Chapter 1. Temperature and Thermal Equation of State
This chapter reviews the major events that led to the development of the physical quantity heat (second half of the 18th century), and then to the understanding that heat and work are not state functions and the introduction of internal energy and entropy as state functions (second half of the 19th century).
Steam engines, which appeared already in the 18th century, played a major role in this development. However, the development of the quantity heat and the advent of calorimetry in the 18th century were not associated with the development of steam engines. Sadi Carnot proposed the theory of the ideal steam engine in 1824; this served as the starting point for the development of classical thermodynamics. We examine the main stages in this development.
The laws of thermodynamics appeared from the operation of an ideal steam engine under the Carnot cycle. However I skip these calculations and start with the expressions for the first and second laws in differential form, paying attention to important things necessary for the next chapters. The features of the idealization introduced by Carnot, related to the introduction of equilibrium and reversible processes, are discussed separately. It is important to understand the connection of integrals from the differential forms of thermodynamics laws with conceptual models for the behavior of substances.
The main obstacle to the study of thermodynamics is the attempt to obtain a visual answer to the question ‘what is entropy?’ immediately upon examining the second law. This often leads to statements about the shortcomings of thermodynamics, which often corrections without studying thermodynamics. I give the example of the Cartesians, who immediately declared Newton’s gravitational force an ‘occult force’ and tried to find their solution without studying Newtonian mechanics — let us not follow such a way.
- Steam engines
- Heat and calorimetry
- Main stages in the development of thermodynamics
- First and second laws of thermodynamics
- Equilibrium and reversible processes
Steam engines
Let us start with a telling quote from Sadi Carnot, from his only work, ‘Reflections on the Motive Power of Fire and on Machines Fitted to Develop that Power‘ (1824):
‘Iron and heat are, as we know, the supporters, the bases, of the mechanic arts. It is doubtful if there be in England a single industrial establishment of which the existence does not depend on the use of these agents, and which does not freely employ them. To take away to-day from England her steam-engines would be to take away at the same time her coal and iron. It would be to dry up all her sources of wealth, to ruin all on which her prosperity depends, in short, to annihilate that colossal power. The destruction of her navy, which she considers her strongest defence, would perhaps be less fatal.’
Newcomen’s first steam engine was used to pump water from coal mines in England in 1712. This design remained unchanged until James Watt’s improved steam engine design in the 1780s. However, in the 18th century, the existence of steam engines had no impact on the development of the concept of heat, as its operation involved too many interrelated processes:
- Combustion of coal (the concept of energetics of chemical reactions did not exist; the phlogiston theory was still in use);
- Evaporation/boiling of water and condensation of steam (there was no theory of phase transformations);
- Steam acted as the working fluid (the properties of steam were not studied);
- Heating and cooling of the cylinder and boiler (the concepts of heat, temperature, thermal conductivity and heat capacity were in their infancy);
- Comparison of the amount of coal burned with the mass of water raised (the concepts of work and energy had not yet been formed, the law of conservation of energy was absent).
On the other hand, in the 19th century, the necessary theories began to appear, although technology continued to be mainly developed thanks to the intuition and ingenuity of engineers. However, in the 19th century, the ties between engineers and scientists were much closer than today, and it was not uncommon for a single person to be both an engineer and a scientist. Let me mention Henri Poincaré’s excellent book on classical thermodynamics (first edition 1892), which contains a chapter on steam engines. Poincaré demonstrates his talent as an excellent engineer. He professionally examines the efficiency of steam engines taking into account all stages, analyzes the contribution of the Carnot cycle and then discusses the possibilities to increase the efficiency of the entire steam engine from the point of view of thermodynamics.
Heat and calorimetry
The concept of heat emerged from experiments on mixing warm and cold liquids. Georg Richmann, at the St. Petersburg Academy, published a rule for mixing hot and cold quantities of water in 1750:
where m1 and m2 are the masses of the water parts to be mixed, and t1 and t2 are their initial temperatures. The concept of heat appeared later in the works of Joseph Black, Johann Wilcke, and others. It was shown that Richmann’s law does not apply when mixing hot water and cold mercury, nor when mixing warm water and ice. Let us begin with the latter, since this effect more clearly demonstrates that the release / takeover of heat can occur at a constant temperature. The effect of constant temperature during the melting of ice was noted before Black, but it was he who introduced the concept of latent heat, that is, the heat of a phase transition. It was then established that latent heat is associated with all phase transitions known at that time.
The next step involved the concept of heat capacity. Different liquids required different amounts of heat to mix until thermal equilibrium was reached. Richmann’s law was then generalized using heat capacities:
where the specific heat capacities C1 and C2 are additionally introduced. They are the same by mixing water, yielding Richmann’s law. They are different by mixing mercury and water, and in this case, it is impossible to describe the observed results without them.
For practical use of the equation above, the unit of heat was introduced. One calorie was defined as the heat required to heat one gram of water by one degree Celsius; thus, the specific heat of water was equal to one. This made it possible to determine the heat capacities of other substances and latent heat. Antoine Lavoisier and Pierre-Simon Laplace proposed a general purpose calorimeter in 1780 (the ice calorimeter), which was widely used to measure heat in the 19th century, including the heat released during combustion.
The ideal measuring instrument, corresponding to the ice calorimeter, is ice at the melting point of 0ºC, within which the substance cools from its initial temperature to 0ºC. The amount of heat generated is equal to the specific heat of fusion of the ice multiplied by the mass of water formed. It is also possible to study a combustion reaction in a vessel within the ice. In this case, the heat of combustion is derived from the mass of water formed. Due to numerous technical difficulties, the accuracy of the ice calorimeter is limited, and other types of calorimeters has been developed. However, an understanding of calorimetry at the level of the ice calorimeter is sufficient for the discussion of classical thermodynamics in this book.
The attempt to define heat as a physical quantity based on what a calorimeter measures corresponds to the caloric theory. It was introduced by Lavoisier and remained in force in the first half of the 19th century. Mathematically, the caloric theory means that heat is a state function. Thus the existence of a caloric equation of state is postulated; for example, in the case of a pure substance, there are two independent variables: Q(T, V). This means that the calorimeter was assumed to measure the change in heat between two states:
Q2 – Q1 = ΔQ
However, research that followed showed that it was impossible to reconcile such a hypothesis with experiments — heat happens not to be a state function after all. The notation ΔQ in modern textbooks and articles is a clear error. The necessary experiments for such a conclusion had already been conducted in the early 19th century (free expansion of a gas, Gay-Lussac, 1807), but due to the lack of alternatives, caloric theory remained.
Note that at these times, in mechanics perpetual motion was rejected, but energy in the modern sense was not conserved. A pendulum would stopped under the influence of friction forces; a body falling to Earth would initially gain speed and vis viva (‘living force’, equivalent to kinetic energy in Leibniz’s terminology), but then, upon impact, vis viva would disappear. This may have been one of the reasons why energy as a physical quantity was absent in mechanics; it only appeared after the first law of thermodynamics.
Main stages in the development of thermodynamics
Many scientists participated in the development of thermodynamics, but I limit myself to a minimal number of names. William Thomson wasn’t yet Lord Kelvin at those times, so he remains as Thomson.
The Carnot cycle within the caloric theory (Carnot, Clapeyron, Thomson)
Sadi Carnot introduced the theory of the ideal heat engine in 1824. His diaries show that he doubted the validity of the caloric theory, but there were no alternatives. Therefore, in his work, he turned to the analogy of falling water — heat passes from high to low temperature, and as it falls, it produces work. Carnot’s work was unnoticed; only Benoît Clapeyron’s paper in 1834 brought attention to Carnot’s ideas.
Sadi Carnot’s genius was in his ability to propose a model of an ideal heat engine, designed to simplify an analysis but still allowing for solid conclusions about the operation of real heat engines. Carnot relied on the impossibility of perpetual motion. This allowed him to prove that 1) the maximum efficiency of a heat engine is independent of the substance of the working fluid; 2) the efficiency of real machines will be lower than that of an ideal one. The second conclusion ultimately leads to the inequality for entropy.
To achieve maximum efficiency, all possible losses had to be eliminated, which in turn led to the disappearance of time in a thermodynamically reversible process; a reversible process was contrasted with an irreversible process. This issue will be discussed separately.
William Thomson was so inspired by the result of the independence of maximum efficiency from the working fluid that in 1848, based on the work of Clapeyron / Carnot, he proposed an absolute temperature scale based on the caloric theory; by the way, in this work he rejected the possibility of converting heat into work.
Mechanical equivalent of heat (Mayer, Joule, Helmholtz)
The calorie, a unit of heat, was introduced in calorimetry. In mechanics, the term ‘work’ appeared in 1826 in the papers of Jean-Victor Poncelet, for example, the work of lifting a load (m g h). Currently, the unit of work is named after one of the participants in the events of that time, James Joule. So today it we talk about a correspondence between the calorie and the joule; the modern value is 1 cal = 4.184 J.
Physician Robert Mayer was the first to demonstrate the equivalence of work and heat in 1842 based on available values of heat capacitances of gases. James Joule, a family brewer, had shown an interest in science and technology from a young age. He has conducted the first experiments to measure the mechanical equivalent of heat in 1843. Joule eventually won Thomson over to his side, which in turn helped him draw the attention of other physicists. Hermann Helmholtz published a paper in 1847 that in general emphasized the conservation of energy (in his paper he has used the term force instead) in various physical processes.
As mentioned, the concept of energy didn’t yet exist. Therefore, all three works dealt with the conservation of force. It could be possible that this very circumstance raised doubts among other physicists. For example, the works of Helmholtz and Joule were rejected by scientific journals, and they published papers at their own expense. Mayer had a hard time getting published in ‘Annalen der Chemie und Pharmacie‘, but after that he published books also at his own expense.
Laws of Thermodynamics (Clausius and Thomson)
The foundations of modern classical thermodynamics were laid by the works of Rudolf Clausius and William Thomson between 1850 and 1865, with Clausius playing the primary role. They formulated two laws of thermodynamics, establishing the existence of two functions of state — internal energy and entropy. At the same time, it was shown that neither heat nor work are functions of state.
In the previous chapter we discussed the thermal equation of state that, for a pure substance, relates temperature, pressure, and volume. The first law of thermodynamics introduces a new state function, internal energy (U), which is associated with a caloric equation of state. The second law introduces a state function, entropy (S), and it turns out that the existing thermal and caloric equations of state are sufficient to determine it. At the same time, neither heat nor work can be attributed to the state of a substance. They appear during the transition of a substance from one state to another, but they are not present in the state of a substance.
First and second laws of thermodynamics
Let us start with the familiar Carnot drawing:
Carnot’s idealization for finding maximum efficiency requires that the temperature and pressure in the working fluid remain uniform and that no temperature or pressure gradients arise during the process. Furthermore, bodies A and B act as ideal heat sources — their temperatures remain constant during heat exchange.
Ideal processes in the Carnot cycle are called equilibrium (quasi-static) or reversible processes. An equilibrium process considers only the state of the working fluid; the temperature and pressure remain uniform throughout the process. The term reversible process is more demanding; it encompasses the entire system, including ideal heat sources A and B. It is assumed that after a reversible process is performed in one direction and then in the opposite direction, the same initial state of the entire system, including the states of the ideal heat sources, is achieved.
It should be clear that equilibrium / reversible processes are not feasible in practice, because piston movement in either direction requires a pressure difference, and heat transfer requires a temperature difference. This will inevitably lead to temperature and pressure gradients in the working fluid. Such processes will be considered non-equilibrium, and there is a proof that the efficiency in this case will be less than that by the ideal processes. The idealization associated with equilibrium and reversible processes and the difficulties to understand them are discussed in more detail in the next section.
The first law of thermodynamics relates changes in internal energy to heat (Q) and work (W). According to IUPAC recommendations, heat and work are considered positive when they increase the internal energy of a substance, so the first law at present looks as follows:
dU = dQ + dW
In older textbooks, there could be a minus sign before work, as there was a different convention for the sign of work. This minus sign has now been incorporated into the expression for work:
dW = −pexdV
In general, the work done by a working fluid is calculated using the external pressure on the piston (pex), since in a non-equilibrium process, a pressure field arises within the working fluid. The above notation allows for the correct calculation of the work performed by or on the working fluid in a non-equilibrium process. In an equilibrium process, the pressure above the piston is equal to the uniform pressure within the substance.
Heat and work differentials are inexact and therefore sometimes they are written in a slightly different form, although in mathematics, they are just as infinitesimal as other differentials. The difference arises during integration — the integral depends on the path, so the value of the integral cannot be written as the difference between the initial and final states:
ΔU = Q + W
Also, the closed contour integral for work and heat will not be equal to zero, unlike to the internal energy:
0 = Q + W
This equation corresponds to the completion of one Carnot cycle, when the state of the working fluid has returned to its initial state. The internal energy is a function of state, and therefore the difference is zero; the state is the same. At the same time, heat is converted into work or work into heat — the closed contour integral for work and heat is not zero, which again emphasizes the impossibility of writing the sign of Δ in this case.
Entropy is related to heat by the second law of thermodynamics:
| Reversible process | |
| Irreversible process |
The both equations belong to the definition of entropy, and in this sense, they contain a hidden answer to the question of what entropy is. The meaning of the introduced state function, entropy, can only be achieved by working examples with these equations out; this is done in the next two chapters. Note that the equations above are equivalent to those with the Carnot cycle; the only difference is that in the Carnot cycle, integrals are taken over a closed contour.
The second law consists from two parts, and it is important to note that entropy in both parts is the same — it is a property of a substance, that is, a function of state. This is the difference with heat on the right-hand side of equations, since heat is not a property of a substance. Therefore, the inequality sign in the second part refers not to entropy, but to heat. The change in entropy from state 1 to state 2 is the same for reversible and irreversible processes, provided that states 1 and 2 are the same in both processes.
The distinction between a reversible and an irreversible process in the equations of the second law is discussed in more detail in the next section. It should be noted that in the case of an irreversible process, the inequality refers to the temperature of the ideal heat source (Tex), not the temperature of the substance. This allows us to consider irreversible processes in the working fluid with temperature gradients.
In the next chapter, ‘Thermodynamic Properties of Substances‘, we switch to functions of state only, to the fundamental equation of thermodynamics, which combines the first and second laws. This allow us to relate entropy of a substance entropy to experiments being conducted, thereby solving the problem of coordination: the second law defines entropy, and the connection with experiments in the next chapter show what can be considered as a measurement of entropy. After that in the chapter, ‘Clausius Inequality as an Equilibrium Criterion‘, we discuss examples of using the second part of the second law, as the fundamental thermodynamic inequality. This produces a criterion for a spontaneous process and simultaneously a criterion for establishing equilibrium.
Equilibrium and reversible processes
Let us start with an expression for the maximum work performed by the working fluid (A = −W). This is the integral from the thermal equation of state of the working fluid p(V, T), which relates the internal pressure (p) to the temperature (T) and volume (V):
Let us search for the necessary idealization to form of a conceptual model that corresponds to such an integral. During actual piston motion, temperature and pressure gradients arise in the working fluid. The equation above however assumes uniform pressure and temperature; thus, the first step of the idealization involves neglecting temperature and pressure gradients. This could be imagined as very slow piston motion, when the temperature and pressure of the working fluid remain almost uniform.
The second step of idealization concerns the external pressure above the piston, since the expression for work must contain this pressure (pex). The integral above for maximum work requires that the uniform pressure of the working fluid remains equal to the external pressure throughout the process. At first glance, this appears to be a contradiction. On one hand, idealization requires mechanical equilibrium of the working fluid (internal pressure equals external pressure); on the other hand, the actual process requires a difference between the two pressures; only then can the piston spontaneously move.
At this point, the concept of a quasi-static process is introduced, where the external pressure changes in very small increments. Usually a mountain of sand above a piston is imagined; we then remove or add a grain of sand during the expansion or compression process. The transition to the limit occurs when the grain size approaches the infinitesimal, giving us the desired integral above. Now volume and pressure of the working fluid are uniform and related by the thermal equation of state and in the limit, the internal and external pressures are equal.
A common objection is that such a limit transition makes the process impossible, since it is assumed that for a transition from one equilibrium state to another requires, equilibrium must be violated. However, this argument confuses the conceptual model with the world. No one argues that for a real process to take place, it is necessary to disrupt the state of equilibrium. At the same time, nothing prevents us from computation of the integral above and imagining precisely this change in a conceptual model. In this sense, the equilibrium process is not spontaneous; it does not occur by itself; the equilibrium process is associated with the power of thought required to take the integral.
In the case of an isothermal process, additional consideration of heat exchange is required, since according to the first law of thermodynamics, the movement of the piston is associated with heat exchange (Q is the amount of heat, ΔU is the change in internal energy):
In the case of an adiabatic process Q = 0 (the working fluid is disconnected from the heater or refrigerator), but in the case of an isothermal process Q ≠ 0 is required.
First, a few words about the ideal heat source. These objects are also products of idealization — it requires imagining an object with given temperatures so large that the transfer or reception of energy in the form of heat does not change its temperature. Thus, connecting the working fluid to one of these objects allows for an isothermal process, where, despite heat exchange, the temperature of the heater and, then, the working fluid remains unchanged.
It should be noted that not every quasi-static process is reversible — thermodynamic reversibility requires a quasi-static process, but this is not sufficient. The concept of thermodynamic reversibility includes not only the state of the working fluid but also the state of all other objects in the system. The idea of a reversible process is that executing the Carnot cycle, first in one direction and then in the opposite direction, should return all objects in the system to their initial state.
This could be only possible if the temperature of the working fluid during the isothermal process is equal to the temperature of the heater. This is due to the relationship between the amount of heat and the change in entropy in an isothermal process: Q = TΔS. Formally, this equation must include the temperature of the heater, and if it differs from the temperature of the working fluid, this will lead to an irreversible increase in the entropy of the entire system. On the other hand, heat exchange between the heater and the working fluid in the real world requires a temperature difference, since heat in a spontaneous process is transferred only from a hotter to a colder body. If two bodies have the same temperatures, they are in a state of thermal equilibrium, and the heat flow between them is zero.
In the case of an isothermal process, a limit transition is again required, similar to that in the case of external pressure. We must begin with a small temperature difference that ensures heat transfer in the desired direction, and then let this difference tend toward an infinitesimal value. In the limit, the temperature of the working fluid is equal to the temperature of the heater, and we obtain the correct amount of heat required for the isothermal process. Thus, when considering reversible processes, we must assume that during an isothermal process, heat exchange occurs between the heater and the working fluid even in the limit when their temperatures are equal.
Let me remind you that Carnot’s problem was to find an expression for the maximum efficiency of a heat engine. The limit transitions discussed above made it possible to solve this problem. For real heat engines, the efficiency will always be lower, and limit transitions allows us to assert that the efficiency obtained in this way is the limiting value when losses are reduced to zero.
Let us consider a pendulum as an example. The concept of a weightless, inextensible thread, as well as replacing the body with a material point, leads to a simple equation of motion in the conceptual model, while the elimination of all friction forces leads to endless oscillations. In this case, idealization is carried out in the spirit of a limit transition. Carnot made a similar transition when constructing an ideal heat engine. The only difference is that Carnot’s limit transitions leads to the absence of time in the final equations, so conceptual motion in reversible processes in the Carnot cycle requires the power of thought to compute the integrals.
Next: Chapter 3. Thermodynamic Properties of Substances
References
Sadi Carnot, Reflections on the Motive Power of Fire and on Machines Fitted to Develop that Power, 1897 (in French in 1824).
H. Poincaré, Termodynamique, 1908 (first edition 1892). I have read Russian Translation: H. Poincaré, Thermodynamics, 2005.
P. S. Kudryavtsev, History of Physics, v. 1, From Antiquity to Mendeleev (in Russian), 1956.
I. R. Krichevskii, The concept and fundamentals of thermodynamics (in Russian), 1970, Chapter III, Heat.
Ya. M. Gelfer, Laws of conservation (in Russian), 1967. Chapter one and three.
Ya. M. Gelfer, History and methodology of thermodynamics and statistical physics (in Russian), 1981. Part one and two.
Additional information
Henri Poincaré: Thermodynamics : A Brief Summary of the Book. A view of classical thermodynamics of the time through the eyes of a renowned physicist. At the end are calculations with the incomplete heat differential dQ(V, p).
Reversible processes in classical thermodynamics : Reversible processes as an idealization in the spirit of limit transition. Mental models in the Carnot cycle corresponding to the calculated integrals in isothermal and adiabatic processes are considered.
Reversible Heat Transfer Between Two Bars with Different Temperatures : The process of heat transfer between two bars with different temperatures is considered. The procedure for reversible heat transfer proposed by the mathematician Zorich is discussed. Zorich’s idea is shown to be effective.
Discussion