Download The Third Law of Thermodynamics and more Lecture notes Thermodynamics in PDF only on Docsity!
CHAPTER 3
THE THIRD LAW OF THERMODYNAMICS^1
In sharp contrast to the first two laws, the third law of thermodynamics can be characterized by diverse expression^2 , disputed descent, and questioned authority.^3 Since first advanced by Nernst^4 in 1906 as the Heat Theorem, its thermodynamic status has been controversial; its usefulness, however, is unquestioned.
3.1 THE HEAT THEOREM
The Heat Theorem was first proposed as an empirical generalization based on the temperature dependence of the internal energy change, (^) ∆ U , and the Helmholtz free energy change, (^) ∆ A , for chemical reactions involving condensed phases. As the absolute temperature, T , approaches zero, ∆ U^ and ∆ A^ by definition become equal, but The Heat Theorem stated that d ∆ U /d T and d ∆ A /d T also approach zero. These derivatives are ∆ C v and - ∆ S respectively. The statement that ∆ C v equals zero would attract little attention today in view of the abundance of experimental and theoretical evidence showing that the heat capacities of condensed phases approach zero as zero absolute temperature is approached. However, even today the controversial and enigmatic aspect of The Heat Theorem is the equivalent statement
(^1) Most of this chapter is taken from B.G. Kyle, Chem. Eng. Ed ., 28(3) , 176
(1994).
(^2) For a sampling of expressions see E. M. Loebl, J. Chem. Educ ., 37 , 361
(1960).
(^3) For extreme positions see E. D. Eastman, Chem. Rev. , 18 , 257 (1936).
(^4) All of Nernst's work in this area is covered in W. Nernst, The New Heat
Theorem ; Dutton: New York, 1926.
S = 0
T 0
lim→
In 1912 Nernst offered a proof that the unattainability of zero absolute temperature was dictated by the second law of thermodynamics and was able to show that Eq. (3-1) follows from the unattainability principle. The latter result seems undisputed, but Nernst was unable to convince his contemporaries of the thermodynamic grounding of the unattainability principle. Many years of low-temperature research have firmly established the credibility of the unattainability principle and as a result it has been proposed as the third law of thermodynamics. This proposal has the merit of having all three laws expressed in phenomenological language and, of course, it leads to the useful result stated in Eq. (3-1). As a matter of convenience, it is possible to express ∆ S for a process under consideration in terms of entropies of formation of participating species because in such a calculation there is a cancellation of the entropies of the constituent elements. For this reason the entropy of an element may be assigned any value. According to Eq. (3-1), at zero absolute temperature the entropy changes for formation reactions will be zero and it is convenient to set elemental entropies equal to zero as recommended by Lewis and Randall.^5 This results in the familiar statement that the entropy of every perfect crystalline substance can be taken as zero at zero absolute temperature and is, of course, the convention employed in the determination of "absolute" entropies.
3.2 CONFORMANCES, EXCEPTIONS, AND INTERPRETATIONS
Undoubtedly the most convincing confirmation of the Heat Theorem involved the calculation of absolute entropies from calorimetric measurements on pure substances which were then used to calculate entropy changes for chemical reactions. These calculated values were in agreement with entropy changes determined from the temperature dependence of experimentally measured equilibrium constants.^6 Later, it was shown through the use of
(^5) G. N. Lewis and M. Randall, Thermodynamics and the Free Energy of
Chemical Substances , McGraw-Hill, New York, 1923, Chap. 31.
(^6) Other successful applications of the third law are given by K. Denbigh, The
Principles of Chemical Equilibrium , 3rd ed., Cambridge University Press, Cambridge, 1971, Chap. 13.
entropy and disorder in a spatial sense. Unfortunately, there are other systems conforming to the Heat Theorem that place a strain on this interpretation. We now examine these systems. Measurements of phase equilibrium data for pure substances show that the slope of the solid-vapor coexistence curve for many substances and the slope of the liquid-vapor coexistence curves for 4 He and 3 He approach zero as zero absolute temperature is approached.^9 From the Clapeyron equation,
dP dT
S
v
and the fact that ∆ v is finite, we obtain the result that (^) ∆ S = 0 and conclude that Eq. (3-1) applies to these phase changes. Both helium isotopes remain liquid under their own vapor pressure down to zero Kelvin and both require a pressure considerably higher than their vapor pressures in order to form a solid phase and the appropriate calculations show^10 that Eq. (3-1) also applies to the solid- liquid phase transition. Thus, if the Lewis and Randall convention is used, the entropies of pure liquids and vapors also have zero entropies. While it may be possible to argue that these systems are nondegenerate in their lowest energy state, the simple picture of zero entropy corresponding to perfect order does not seem appropriate, at least in a physical sense. The interpretation is further strained when the behavior of glasses in the low-temperature limit is considered. The Maxwell relationship
T
=- V
P
S
T P
together with Eq. (3-1) leads to
= 0 T
V
T (^0) P
lim→ (3-5)
Thermal coefficients of expansion for many substances have been measured at
(^9) J. A. Beattie and I. Oppenheim, Principles of Thermodynamics ; Elsevier,
Amsterdam, 1979, Chap. 11.
(^10) ibid.
temperatures approaching absolute zero. As expected, Eq. (3-5) is obeyed by crystalline solids, but one may be surprised to learn that it is also obeyed by glasses.^11 Here a microscopic physical interpretation hardly seems possible. Systems comprised of liquids, vapors, and glasses strain to the breaking point the putative association of zero entropy with perfect spatial order. These are the systems that prompt us to ask Is there a microscopic physical interpretation of the Heat Theorem applicable to all systems? One could argue that the association of entropy with spatial order is naive and that (^) Ω 0 = 1 only means that the system is nondegenerate. For example, both Fermi-Dirac and Bose-Einstein gases have been shown to be nondegenerate^12 and therefore have S o = 0. In the case of crystalline solids Ωo = 1 can be interpreted physically as spatial order, but no such microscopic description of the gases in physical terms is possible. Instead Ωo can only be seen as a logical construct that allows a mathematical treatment of the system. The answer to the question is No! only an explanation in logical terms is possible. This is yet another instance of our inability to obtain a microscopic view of entropy in anything other than logical terms. If there is no physical microscopic interpretation of the Heat Theorem, then what is the basis for its existence? As will be shown below, the answer is that Eq. (3-1) is dictated by the logical structure of thermodynamics.
3.3 THE CLASSICAL THERMODYNAMIC VIEW
The absolute temperature scale is defined in terms of the performance of a Carnot engine
|Q |
=|Q|
T
T
2
1 2
where Q 2 is the input heat at T 2 and Q 1 is rejected heat at T 1. Instrumental in the derivation of Eq. (3-6) is a second-law statement such as It is impossible to
(^11) G. K. White, Cryogenics , 4 , 2 (1964).
(^12) A. H. Wilson, Thermodynamics and Statistical Mechanics , Cambridge
University Press, Cambridge, 1957, Chap. 6.
S 0 β^ - S 0 α≤^0 (3-10)
Considering the reverse process that proceeds from T ″ to T ′ = 0, we can, in the same manner, show that it is necessary for
S 0 β^ - S 0 α≥^0 (3-11)
These two inequalities can be satisfied only when S 0 β^ = S 0 α^ and it is seen that Eq. (3-1) follows from the unattainability principle. Thus, Eq. (3-1) arises from the second law and is needed to preserve the logical structure of thermodynamics; a third law is therefore unnecessary.
3.4 ADDRESSING PREVIOUS ARGUMENTS
Two types of arguments found in the literature should be addressed: those that attempt to show that the attainment of zero absolute temperature is not prohibited by the second law and those that attempt to show that existence of a reservoir at zero absolute temperature does not threaten the second law. By applying the mathematical formalism of thermodynamics down to and including zero absolute temperature, it has been shown that this temperature can be reached in a finite number of steps^15 or that the work required to reduce a system^16 to this temperature is finite.^17 As shown in Appendix 3A, the mathematical formalism is such that the use of T = 0 as a lower limit of integration gives the appearance of being permissible. A similar condition probably obtains for these arguments which, despite their apparent cogency, are incomplete because the possibility that the existence of a reservoir at zero absolute temperature might pose a threat to the second law was not examined. Nernst's proof that the unattainability principle is required by the second law was based on the argument that if a reservoir at zero absolute temperature existed, it would be possible to operate a Carnot engine using this reservoir to
(^15) F. E. Simon, Low Temperature Physics , Academic Press, London, 1952,
Chap. 1.
(^16) The heat capacity of the system was assumed to follow Eq.(3-12) of
Appendix 3A.
(^17) E.M.Loebl, op. cit.
convert heat taken in at a higher temperature completely into work. This is essentially the argument presented here. The two most damaging objections against this position were based on possible operating difficulties associated with the Carnot cycle.^18 19 The first objection calls into question the possibility of carrying out an isothermal process at zero absolute temperature because the effects of heat leaks and frictional heat are much more pronounced at this extreme. This is an objection of degree rather than principle and should carry no weight when it is recognized that the logical structure constituting thermodynamics rests on such idealizations as reversibility, isothermality, and adiabaticity. As these idealizations can never be realized in the physical world, it seems pointless to argue that they would be more difficult to achieve at low temperature. The second objection points to the ambiguity associated with the isothermal step in the Carnot cycle presumed to occur at zero absolute temperature. Because no heat is rejected, this step would be adiabatic as well as isothermal but it would not necessarily be isentropic for it can only be said that the entropy change is 0/0. It has been argued that when a system attempting to follow a Carnot cycle reaches zero absolute temperature the second law is not threatened because there is no assurance that the system would take the isothermal path and complete the cycle rather than take the adiabatic path and return to a previous state. The emphasis here is misplaced! Because a single violation would vitiate the second law, concern should be directed to the possibility, no matter how small, that the system would take the isothermal path. There is no assurance that this would not occur and therefore the unattainability principle is needed. Both of these inoperability objections seem to demand a premature reality check. It could be argued that the Carnot engine is simply a concept that is part of the logical, mathematical formalism of thermodynamics and it is rather the
(^18) P. S. Epstein, Textbook of Thermodynamics , Wiley, New York, 1937, Chap.
(^19) A. B. Pippard, The Elements of Classical Thermodynamics , Cambridge
University Press, London, 1957, Chap. 5.
(^20) M. L. Boas, Am. J. Phys ., 28 , 675 (1960).
"For any isothermal process involving only phases in internal equilibrium or, alternatively, if any phase is in frozen metastable equilibrium, provided the process does not disturb this frozen equilibrium, limT→ 0 ∆ S = 0."
Simon assigned unquestioned thermodynamic status to Eq. (3-1) and pointed out that the restrictions made explicit in his statement are implicitly made in applying any other thermodynamic relationship. The question is not whether Eq. (3-1) is valid, but rather whether the application of thermodynamics to a particular system is valid. Valid thermodynamic systems must exist in equilibrium states and thus be capable of undergoing reversible processes. As Eq. (3-1) is applied only under the most stringent conditions where "frozen-in," nonequilibrium states are not unexpected, it is natural that it will not seem to possess the unexceptional status accorded to the other laws and relations of thermodynamics. However, this is a problem in the application of thermodynamics and should not call the validity of Eq. (3-1) into question. Because of the widespread use of the Lewis and Randall convention leading to the convenience of "absolute" entropies and because of the remarkable success in calculating these values via the methods of quantum statistical mechanics, we are tempted to regard entropy as an intrinsic property of matter and thereby seek a physical microscopic interpretation. However, we have seen for the case of liquids, gases, and glasses that this is not a fruitful approach. Equation (3-1) is the most general statement and has been shown to be simply a necessary logical statement. This suggests the view that entropy is merely a defined state function embedded in the logical-mathematical structure of thermodynamics. Thus, it seems appropriate that quantum statistical mechanics yields a representation of entropy in logical rather than physical terms. Because classical thermodynamics neither provides nor requires physical visualization of its functions, entropy is no less useful for want of a microscopic physical interpretation. While this view does not provide the physical insight available through a microscopic interpretation, it is at least free of contradictions.
3.6 ABSOLUTE ENTROPIES
If the requisite calorimetric data are available, the entropy of an ideal gas can, in general, be calculated from
dT T
+ C (g) T
dT+ h T
+ C (l) T
dT+ h T
+ C (c) S =S P
T
V T
T P V F T
T P F
0
o V
V
F
F
where C P(c), C P(l), and C P(g) are the heat capacities of the crystalline, liquid, and gas phases, ∆ h Fand ∆ h Vare the latent heats of fusion and vaporization, and T F and T V are the corresponding phase transition temperatures. We have already seen that when elements are assigned zero entropy at zero absolute temperature, the third law specifies the entropy of any substance, S o, to be zero at this condition. This convention allows values of "absolute" entropy to be calculated from calorimetric data via the above equation. This is a convenience and not a necessity. In fact, in using this convention the isotope effect has been ignored. Most elements consist of a mixture of isotopes. From the microscopic point of view, there will be many possible arrangements of these sets of distinguishable particles on the crystal lattice and therefore (^) Ωo will be significantly greater than unity. From a macroscopic point of view, we would simply say that there is an entropy of mixing to be considered. Because the isotopic composition of the elements is now well known, the isotopic effect could be dealt with quantitatively. However, as a practical matter it is not necessary to do so because the isotopic composition is expected to remain constant for processes of thermodynamic interest and therefore any correction for this effect would cancel on determining an entropy change. Using quantum statistical mechanics the Sackur-Tetrode equation, Eq. (2- 21), for the entropy of a monatomic ideal gas was derived in Sec. 2.3.2.
] 2
)+^5
h
(^2 mkT 2
+^3
N
S =Nk[ V 2 ln ln^ π (2-21)
This equation yields absolute entropy values in excellent agreement with calorimetric values. Since most of the noble gases are isotopic mixtures, this situation must be considered in applying Eq. (2-21). As Eq. (2-21) shows a dependence of entropy on particle mass, the entropy of each isotope can be calculated separately and a weighted average determined for the mixture. Consistent with the convention used for the calorimetric absolute entropy, the entropy of mixing the isotopes is ignored. Thus, "absolute" entropies are not absolute. While the concept of absolute entropy derives from the third law and the Sackur-Tetrode equation yields excellent values of absolute entropy,it will be
APPENDIX 3A
ENTROPY CHANGES NEAR ABSOLUTE ZERO
For a heat capacity of the form
C= T β
α α β^ (3-12)
the entropy change between an extremely low temperature, (^) ε, and the temperature of interest, T , is
dT= (T - ) T
S-S =T T ε β
α β α β β ε
and the limit as (^) ε approaches zero is
dT T
= C
(S-S )= T
T (^0 )
ε β ε β
α lim (3-14)
which is seen to be the result obtained by using T^ = 0 as the lower limit of integration.
