1.4. Clausius Inequality as Equilibrium Criterion

Content

Previous: Chapter 3. Thermodynamic Properties of Substances

Carnot’s idealization allows us to find the maximum efficiency of a heat engine; in a real heat engine, the efficiency is less than the maximum value. This fact leads to the Clausius inequality, which expresses this fact by means of a state function entropy. In classical thermodynamics, the Clausius inequality defines the criterion for a spontaneous process and simultaneously the criterion for equilibrium — the state at which the spontaneous process terminates.

We examine the Clausius inequality for an isolated system and consider the simplest example of heat death. The Clausius inequality with the first law yields the fundamental inequality of thermodynamics. A simple example of using the fundamental inequality is given: the ideal gas reaching the state of equilibrium. These simple examples demonstrate how non-equilibrium states are treated in classical thermodynamics.

In conclusion, the non-equilibrium states in the examples discussed are generalized to more complex non-equilibrium states. Also there is a discussion of the point mentioned in the chapter related to temperature. The definition of temperature by itself requires the condition of achieving thermal equilibrium, so the equilibrium criterion based on the Clausius inequality should be connected with this fact. The concepts of local and global equilibrium allows us to solve this problem.

• Clausius inequality for an isolated system
• The simplest example of heat death
• The fundamental inequality of thermodynamics
• Local and global equilibrium. Construction of thermodynamics

Clausius inequality for an isolated system

The inequality in the second law of thermodynamics (the Clausius inequality) poses particular difficulties to study classical thermodynamics. Let us write down the inequality only:

dS > dQ/T_ex

In an isolated system with no energy exchange with its surroundings, there is no heat exchange and no work is performed. Thus, the internal energy and volume remain constant, and the Clausius inequality looks like as follows:

(dS)_U,V > 0

Subscripts remind us that internal energy and volume are constant. This is now the criterion for a spontaneous process in an isolated system — the entropy of the isolated system can only increase. When the equilibrium state is reached, the spontaneous process stops, and the entropy reaches its maximum value constrained by the specified values ​​of internal energy and volume.

At this point, a certain logical problem arises. The assertion that entropy reaches the maximum value at the equilibrium state implies that classical thermodynamics can deal with non-equilibrium states. On the other hand, it is often said that the equality in the second law can only be used for equilibrium processes — that is, classical thermodynamics cannot deal with the entropy of non-equilibrium states.

This question hinges on the ambiguity of the term ‘non-equilibrium state’. We begin with simple examples that show that classical thermodynamics at least provides tools for dealing with some non-equilibrium states. This assertion is then generalized to more complex non-equilibrium states.

The simplest example of heat death

Let us begin with a simple example, where an isolated system consists of two subsystems with identical heat capacity (C_V,1 = C_V,2), independent of temperature. Two subsystems are separated from each other by a fixed wall — index 0 characterizes the initial temperature values:

$$T_{1,0}\mid T_{2,0}$$

The subsystem volume does not change, and the only possibility of change is related to temperature: the wall allows heat to pass through. Thus, in the equations for internal energy and entropy, only the temperature part related to the heat capacity remains (dV=0). Let us assume that the transfer of internal energy as heat occurs extremely slowly and hence there is no temperature gradients in either subsystem. In other words, both subsystems are in a state of local equilibrium with uniform temperatures.

The whole system, on the other hand, is not in a state of equilibrium, since energy exchange between subsystems is possible. The task now is to consider all possible states in the whole system. The total energy of the system is conserved: U = U₁ + U₂ = const; hence in the case of the equal heat capacities, a simple relation between the temperatures is obtained:

$$d{U}_1=–d{U}_2$$ $$C_{V}d{T}_1=–C_{V}d{T}_2$$ $$T_1=T_{1,0}+x\mid T_2=T_{2,0}–x$$

The variable x shows the changes in temperature, the increase in temperature in one subsystem is equal to the decrease in temperature in the other. The total internal energy does not depend on x, since the last relation is obtained from the total internal energy being constant.

At the same time, the total entropy depends on the values of the current temperatures in the subsystems, that is, on the value of x. In the equation below, the integral of entropy over temperature is taken, and S₀ denotes the total entropy in the initial state:

$$S=S_1+S_2=S_0+C_{V}ln\frac{T_1}{T_{1,0}}+C_{V}ln\frac{T_2}{T_{2,0}}$$ $$S=S_0+C_{V}ln\frac{T_{1,0}+x}{T_{1,0}}+C_{V}ln\frac{T_{2,0}–x}{T_{2,0}}$$

Let us plot this function – the dependence of the total entropy on x. Below this is plotted for the case of initial temperatures of 300 K and 400 K, with S₀=0, C_V=1 with Gnuplot (the script is in Appendix).

The change in temperature is shown on the right axis of the ordinate: T₁ = 300 + x, T₂ = 400 – x, the change in entropy on the left. The entropy reaches the maximum value at x = 50 when the temperatures of the subsystems become equal as expected.

The maximum of entropy corresponds to zero of the entropy derivative over x, which allows us to get the answer in a general form:

$$dS/dx=0,x=\left(T_{2,0}–T_{1,0}\right)/2$$ $$T_{2,eq}=T_{1,eq}=\left(T_{2,0}+T_{1,0}\right)/2$$

A non-equilibrium state in this example is related to the case where an isolated system consists of subsystems, each of which has uniform temperatures. The subsystems by themselves are in the equilibrium state, but they are not in equilibrium with each other.

The example shows the use of the entropy of an isolated system to compute the equilibrium state. The red curve in the figure corresponds to all possible non-equilibrium states of the system, and the maximum value on this curve corresponds to the global equilibrium. The Clausius inequality also sets the direction of a spontaneous process – the entropy of an isolated system can only increase.

In the language of heat engines, the inequality reflects the asymmetry of work and heat. All work can be converted into heat, but only a part of heat can be converted into work. In an isolated system, which includes both a heat engine and surrounding bodies, the possibility to perform work is eventually lost as energy is dissipated.

This example leads to a pessimistic interpretation in the form of the heat death of the universe. After the introduction of the second law, Rudolf Clausius did not miss the opportunity to transfer results to the entire universe: ‘The energy of the world is constant. The entropy of the world tends toward a maximum.’ This is a classic example of extrapolationism, when the results of a study of a particular case (a heat engine) are immediately and without hesitation applied to the entire universe.

The fundamental inequality of thermodynamics

The combination of the Clausius inequality with the first law gives the fundamental inequality of classical thermodynamics:

dU < T_exdS ‐ p_exdV

The inequality includes external temperature and pressure, which are fixed and therefore no longer functions of the system’s entropy and volume. This makes the fundamental inequality significantly different in meaning as compared with the fundamental equation discussed in the previous chapter. The fundamental equation describes the change in the thermodynamic properties of a substance in the space of independent variables, and the fundamental inequality is a criterion for the spontaneity of a process during evolution of the system to the equilibrium state under given external conditions. Let us again consider Sadi Carnot’s drawing to understand the difference:

Let us consider the following. A substance at uniform temperature and pressure (T₁, p₁) was contained in a cylinder. The cylinder was then connected to a heat source with temperature T_ex, and the external pressure was changed to p_ex. At this point, the spontaneous transition of the substance from its initial to its final state begins. To examine this process, it is convenient to move all terms to the left-hand side of the inequality:

dU ‐ T_exdS + p_exdV < 0

This defines a function that can only decrease in a spontaneous process and that reaches the minimal value in the final equilibrium state. Thus, the criterion for a spontaneous process is simultaneously the criterion to reach the final equilibrium state.

In classical thermodynamics, the final equilibrium state corresponds to uniform temperature and pressure — all gradients vanish. Therefore, it is possible to simplify the problem and restrict the consideration to substance states with uniform temperature and pressure. The problem formally now is as follows: considering all possible states of the substance with uniform temperatures and pressures, find the equilibrium state that meets the external conditions.

The answer is intuitively clear, but let us see formally that the inequality above is in agreement with intuition — that the inequality leads to the establishment of thermal and mechanical equilibrium. In parallel, this example demonstrates the relativity of the concept of equilibrium — a state with uniform temperature and pressure can correspond to the equilibrium state, but as in this case, it also could correspond to a non-equilibrium state with respect to external conditions.

Under the constraints of uniform temperature and pressure, dU in the inequality can be replaced by the expression from the fundamental equation, since it can be always applied to describe changes between states with uniform temperature and pressure. This yields the following expression:

(T ‐ T_ex)dS + (p_ex ‐ p)dV < 0

It should now be clearer that the minimum of the function on the left is achieved when the substance reaches a temperature and pressure equal to the external pressure and temperature, as expected the inequality leads to the criterion of thermal and mechanical equilibrium.

To visualize this, the values of the left-hand side is computed below for one mole of the ideal gas: pV = RT (R is the universal gas constant). The simplest expressions for the calculation is achieved in variables temperature and volume, so first the transformation is performed. Let us write down the entropy as a function of temperature and volume:

$$dS=\frac{C_V}{T}dT+{\left(\frac{\partial p}{\partial T}\right)}_{V}dV=\frac{C_V}{T}dT+\frac{R}{V}dV$$

The first part of the equation is valid for any substance; C_V is the heat capacity at constant volume, which will be assumed independent of temperature. The second part is obtained by calculating the derivative for the ideal gas equation of state. Substituting this expression into the inequality plus using the equation of state of an ideal gas plus simplification leads to the final expression of the inequality in the case of one mole of the ideal gas:

$$\left(T–T_{ex}\right)\frac{C_V}{T}dT+\left(\frac{R{T}_{ex}}{V_{ex}}–\frac{R{T}_{ex}}{V}\right)dV<0$$

The variables are separated, and thus the integral can be taken over the transition from state (V₁, T₁) to (V₂, T₂); this gives the final expression for the calculation:

$$C_V\left(T_2–T_1\right)–T_{ex}{C}_{V}ln\frac{T_2}{T_1}+\frac{R{T}_{ex}}{V_{ex}}\left(V_2–V_1\right)–{RT}_{ex}ln\frac{V_2}{V_1}<0$$

In this expression, (V₁, T₁) refers to the initial state, (V_ex, T_ex) refers to the external conditions, and (V₂, T₂) means an intermediate state. The inequality says which intermediate states are permissible from the point of view of the laws of thermodynamics during the transition from (V₁, T₁) to (V_ex, T_ex). Also this allows us to find the minimum value of the function on the left side that corresponds to the equilibrium state with respect to the external conditions (V_ex, T_ex). Below is a contour plot for the function on the left-hand side of the inequality when two states, (V₁, T₁) and (V_ex, T_ex), are fixed. Hence this is a function of the intermediate state (V₂, T₂) only (see Gnuplot script in Appendix):

As expected, the function reaches the minimum values at (V_ex, T_ex); this is the final equilibrium state. Positive values ​​of the inequality show that, under these conditions, a spontaneous transition from (V₁, T₁) to this region is thermodynamically impossible. The substance can spontaneously change its state only in the direction to decrease the left-hand side of the inequality. At the same time, the inequality does not allow one to estimate the transition path, nor does it contain information about time. The inequality only allows us to rank possible states during the transition from (V₁, T₁) to (V_ex, T_ex) into thermodynamically permissible and thermodynamically inadmissible, and it also says that the minimum value corresponds to the equilibrium state.

Local and global equilibrium. Construction of thermodynamics

In both examples a subsystem with uniform temperature and pressure was considered. A non-equilibrium state was obtained when two subsystems came into contact or one subsystem came into contact with external conditions. This demonstrates the relativity of the concept of ‘an equilibrium state’ — a subsystem can be in equilibrium on its own, while non-equilibrium is caused by contact with other subsystems or external conditions. In this case, the local equilibrium should be distinguished from the global equilibrium. The global non-equilibrium state of the entire system is broken down into subsystems, each of which is in the local equilibrium. Subsystems in the local equilibrium then exchange energy with each other and with external conditions until the global equilibrium between all subsystems and external conditions is achieved.

In both examples, the entire system can be called discontinuous — it consists of subsystems, each with a uniform temperature. This approach can be generalized to the case of a single system with a temperature field by dividing the original system into infinitely small subsystems. Thus, classical thermodynamics provides means to deal with non-equilibrium states by the local equilibrium principle.

The introduction of local and global equilibrium also solves one of the problems in the construction of thermodynamics. It was already noted that the definition of temperature already requires the assumption of the establishment of thermal equilibrium between the thermometer and the system. Similarly, in the first law was assumed that the mechanical equilibrium is established after the piston’s movement. Though, the latter belongs actually to mechanics.

The local equilibrium principle was used implicitly during the development of thermodynamics, where the Clausius inequality provides the criteria for global equilibrium. Moreover, the criteria for global equilibrium led to results that could not be directly obtained from the local equilibrium principle without the second law. For example, Gibbs’s classic work ‘On the Equilibrium of Heterogeneous Substances‘ (1876-1878) uses the global equilibrium criteria in relation to chemical and phase equilibria. Gibbs’s work forms the framework to discuss the direction of chemical reactions and, therefore, plays an important role in understanding candle combustion.

At the same time, to discuss the arrow of time, we must not forget that the establishment of thermal equilibrium is a prerequisite of classical thermodynamics, just as the establishment of mechanical equilibrium is a prerequisite of continuum mechanics. Thus, strictly speaking, the second law shows the direction toward global equilibrium based on the existence of local equilibrium.

In this light, let me say a few words about axiomatization. There are attempts to simplify the construction of thermodynamics based on the principle of global equilibrium. First, internal energy and entropy are introduced, and then temperature is defined as the derivative of internal energy with respect to entropy: T = (∂U/∂S)_V. For example, see the axiomatization by Lieb and Yngvason in the paper, ‘The Physics and Mathematics of the Second Law of Thermodynamics‘ (1999).

In my view, such construction of thermodynamics is problematic, since it is unclear how to incorporate the temperature field, as well as the pressure field. Without this, the status of the Clausius inequality is unclear, as it presupposes certain tools to deal with non-equilibrium states. I return to these issues in the final chapter of this part, Chapter 6, ‘Entropy of Non-equilibrium States‘, where the relationship between classical and non-equilibrium thermodynamics is also examined.

Next: Chapter 5. Adiabatic Flame Temperature

Appendix

Gnuplot script to produce the figure in the first example:

set terminal png enhanced size 500,400
set output 'fig1.png'
set xlabel 'x'
set ylabel 'S'
set y2label 'T'
set y2tics
set ytics nomirror
set key center bottom
plot [x=0:100] log((300+x)/300)+log((400-x)/400) title 'S', 300+x axis x1y2 title 'T1', 400-x axis x1y2 title 'T2'

Gnuplot script to produce the figure in the second example:

set nokey
set terminal png enhanced size 500,400
set output 'inequality_map.png'
set style textbox noborder
set view map scale 1
set isosamples 200, 200
unset surface
set contour base
set cntrlabel format '%8.3g' font ',7' start 20 interval 50 
set cntrparam levels incremental -24, 6, 18
set style data lines
R = 8.314
Vex = 1
Tex = 300
Cv = 2.5*R
V1 = 1.1
T1 = 320
fun(V2, T2) = Cv*(T2-T1)-Tex*Cv*log(T2/T1)+R*Tex/Vex*(V2-V1)-R*Tex*log(V2/V1)
set label "Vex,Tex" at Vex, Tex center font ',8'
set label "V1,T1" at V1, T1 center font ',8'
set xlabel "V2" 
set xrange [ 0.95 : 1.15 ]
set ylabel "T2" 
set yrange [ 290 : 330 ]
splot fun(x, y) with lines, fun(x, y) with labels boxed

References

Elliott H. Lieb and Jakob Yngvason. The physics and mathematics of the second law of thermodynamics. Physics Reports 310, no. 1 (1999): 1-96.

Additional information

Clausius Inequality in Philosophy and History of Physics: Critique of Uffink’s paper ‘Bluff your way in the second law of thermodynamics‘. Temperature and temperature field in 19th century physics. Continuum mechanics and Clausius Inequality. Physics and mathematics in thermodynamics.

In Russian. Clifford Truesdell and Classical Thermodynamics: A discussion and critique of Truesdell’s position on classical thermodynamics, as the Clausius inequality plays an important role in classical thermodynamics. A comparison with Poincaré thermodynamics.

Discussion

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