H.2 Euler-Maclaur in Sum Formula
3.1 Statement of the Second Law
For a thermodynamic system, there exists a function of state,S, called the entropy.Sis a function of a complete set of extensive state variables that includes the internal energy, U. For allotherextensive variables held fixed,Sis a monotonically increasing function of the internal energyU. For a homogeneous system,Sis an extensive function and its slope
∂S/∂U=1/T, where the positive quantityT is the absolute temperature.2If the system is a composite system,Sis the sum of the entropies of its constituent subsystems.
An isolated system is a chemically closed system for which δQ=0 and δW=0, so dU=0 and U is a constant. Therefore also Q=0, W=0, and U=0. For an isolated system, changes ofSobey the inequality
S≥0, isolated system, allowed changes, (3.1) where the inequality corresponds to a natural irreversible process and the equality corre- sponds to a hypothetical idealized reversible process.
If the entropy of an isolated system is a maximum subject to its internal and external constraints, all natural irreversible processes are forbidden by Eq. (3.1) so the system is in a state of equilibrium. This leads to the following equilibrium criterion:
Entropy criterion for equilibrium: The criterion for an isolated thermodynamic system to be in internal equilibrium is that its total entropy be a maximum with respect to variation of its internal extensive parameters, subject to external constraints and any remaining internal constraints. Isolation constitutes the external constraints of chemical closure, perfect thermal insulation and zero external work, which require the internal energy to be constant.
For example, consider an isolated composite system consisting of two subsystems having different temperatures and separated by an insulating wall (internal constraint). If
2For a homogeneous system, the absolute thermodynamic temperature isdefinedby a partial derivative 1/T: =∂S/∂Uor alternatively byT=∂U/∂S, where all other members of the complete set of extensive variables are held constant. ThusT exists independent of any particular measuring device (thermometer). See Fermi [1, p. 45] for a related discussion in terms of the Carnot cycle.
the wall is then allowed to conduct heat (removal of an internal constraint), the energies of the two systems will change until the temperatures are equalized and a new equilibrium, corresponding to a state of higher entropy, is established.
In Chapter 6 we will discuss the application of this entropy criterion for equilibrium and deduce from it several alternative and useful criteria for equilibrium.
3.1.1 Discussion of the Second Law
The second law of thermodynamics is a postulate. The fact that it is believed to be true is based on extensive experimental testing. It can be rationalized on the basis of statistical mechanics, which of course is based on its own postulates. It can also be derived, as is done in classical thermodynamics for chemically closed systems, from other postulates of Kelvin or Clausius, as stated above. In order to make contact with the historical development of the second law and to derive equations that allow calculation of the entropy, we first digress to apply Eq. (3.1) to a composite system consisting of sources of heat and work.
We consider an isolated composite system having total entropyStotand apply Eq. (3.1) in the form
Stot≥0, isolated system, allowed changes. (3.2) We assume that our composite system consists of a chemically closed system of interest having entropyS, a heat source having entropySs, and a purely mechanical system capable only of exchanging work. By definition, there is no entropy associated with this purely mechanical system, so the total entropy of our composite system is
Stot=S+Ss. (3.3)
The heat source is assumed to be a homogeneous thermodynamic system whose only function is to exchange heat; it does no work, has a fixed number of moles of each chemical component, a temperatureTsand an internal energyUs. Thus dSs=(1/Ts)dUsby definition of the absolute temperature of the heat source. We denote byδQa small amount of heatextractedfrom the source.3From the first law we have−δQ=dUs, so dSs= −δQ/Ts. Thus dStot=dS−δQ/Tsand for infinitesimal changes, Eq. (3.2) becomes
dS≥δQ
Ts, chemically closed system, allowed changes. (3.4) In Eq. (3.4), the term chemically closed system pertains to the system of interest, having entropy S. The inequality pertains to a natural irreversible process and the equality pertains to an idealized reversible process. Thus
dS>δQ
Ts, chemically closed system, natural irreversible changes. (3.5)
3δQis assumed to be so small and the heat source has, by definition, a sufficiently large heat capacity that it remains practically unchanged during this process.
34 THERMAL PHYSICS
For reversible heat flow, which is an idealization that separates irreversible heat flow from forbidden heat flow, Ts can differ only infinitesimally from T, the temperature of the system, so we have
dS=δQ
T , chemically closed system, idealized reversible changes. (3.6) Equations (3.5) and (3.6) are sometimes offered as a statement of the second law, although the distinction betweenTsandTis not always made.4
If our system of interest were simply another heat source capable of no other change, we would have dS=dU/T by definition of its absolute temperature. Then δW=0 so dU=δQfrom the first law and we would have dS=δQ/T. For spontaneous heat conduc- tion, a natural irreversible process, we would need
dStot=dS+dSs=δQ 1
T − 1 Ts
>0, (3.7)
which results inδQ(Ts −T) >0. This means that spontaneous heat conduction, with no other change, occurs only from a higher temperature to a lower temperature, in agreement with our intuition and the postulate of Clausius stated above.
For finite changes, we can integrate Eq. (3.4) to obtain S≥ δQ
Ts, chemically closed system, allowed changes, (3.8) where the equality sign is for a reversible process and requires Ts=T. Our system of interest can do work (on the mechanical subsystem) of amount
W= −U+
δQ, (3.9)
provided that Eq. (3.8) is satisfied. We emphasize that our system of interest is not isolated, so its entropy can be made todecreaseby extracting heat reversibly. Therefore, if a chemically closed system is not isolated, its entropy can increase or decrease, and the process that brings about this change can be either reversible or irreversible, depending on the relationship ofSto
δQ/Tsfor that process.
In classical thermodynamics, one often speaks ofheat reservoirs. A heat reservoir is a heat source with such a large heat capacity that its temperature remains constant.5If the heat source in Eq. (3.8) is replaced by a heat reservoir of temperatureTrfrom which an amount of heatQris extracted, we obtain
S≥ Qr
Tr, chemically closed system, allowed changes. (3.10)
4See the footnote on page 48 of Fermi [1] for further discussion ofTs. Some books [5, 16] write dS> δQ/T which is more restrictive than Eq. (3.5); such an equation applies to a process in which the heat conduction between the heat source and the system of interest is reversible but other processes that take place within the system of interest are irreversible.
5For example, if a heat source has a constant heat capacityCrand an amount of heatQris extracted from it, its temperature would change byTr= −Qr/Cr. For a reservoir,Cris assumed to be so large thatTrcan be made arbitrarily small, and therefore zero for all practical purposes.
If the heat source consists of a number of such reservoirs, Eq. (3.8) becomes
S≥
r
Qr
Tr, chemically closed system, allowed changes (3.11) and Eq. (3.9) is replaced by
W= −U+
r
Qr. (3.12)
If the amounts of heatQrin Eqs. (3.11) and (3.12) are very small, the sums can be replaced by integrals, and the result is essentially the same as Eqs. (3.8) and (3.9).
A system surrounded by perfectly insulating walls requires δQ=0 and is said to be adiabatic. For an adiabatic system, Eq. (3.8) becomes
S≥0, chemically closed adiabatic system, allowed changes. (3.13) But Eq. (3.9) yieldsW= −U, so such a system is not isolated and can still do work. Chan- dler [12, p. 8] states the second law by means of Eq. (3.13) which applies to transformations that are adiabatically accessible, those corresponding to the inequality being irreversible and those corresponding to the equality being reversible.
For a cyclic process, the system returns to its original state after each cycle. SinceSis a state function,S=0 for a cyclic process and Eq. (3.11) becomes
0≥
r
Qr
Tr
, cyclic process, chemically closed system, allowed changes. (3.14) For a continuous distribution of reservoirs,
0≥ δQ Tr
, cyclic process, chemically closed system, allowed changes. (3.15) For an adiabatic cyclic process, δQ=0, so Eq. (3.15) becomes 0 ≥ 0 and compatibility would require the equality sign to hold, consistent with the fact that an adiabatic cyclic process is reversible.