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For a mechanically isolated system kept at constant temperature the Helmholtz free energy never increases.
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For a mechanically isolated system kept at constant temperature the state of equilibrium is the state of minimum Helmholtz free energy. In an infinitesimal reversible transformation it is easily verified that
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dA = -PdV - SdT
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From this follow the relations (1.30) (1.31) which are members of a class of relations known as Maxwell relations. If the function A(V, T) is known, then P and S are calculable by (1.30) and (1.31). As an example of the principle of minimization of free energy, consider a gas in a cylinder kept at constant temperature. A sliding piston divides the total volume V into two parts Vi and V2 , in which the pressures are respectively Pi and P2 . If the piston is released and allowed to slide freely, what is its equilibrium position By the principle just stated the position of the piston must minimize the free energy of the total system. Suppose equilibrium has been established. Then a slight change in the position of the piston should not change the free energy, since it is at a minimum. That is, SA = O. Now A is a function of Vb V2 , and T. Hence
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Since VI + V2 = V remains constant, we must have c5VI = -c5V2 Therefore
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As BVI is arbitrary, its coefficient must vanish. We thus obtain the equilibrium condition
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which becomes, through use of (1.30),
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a result that is intuitively obvious. We consider now the Gibbs potential. Its importance lies with the following theorem.
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For a system kept at constant temperature and pressure the Gibbs potential never increases.
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For a system kept at constant temperature and pressure the state of equilibrium is the state of minimum Gibbs potential.
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is kept constant, then in any transformation
-aA Now specialize the situation further by keeping the pressure constant, thereby making W = P aVo We then have pay + llA ::; 0 llG ::; 0 In a infinitesimal reversible transformation
dG = -SdT + VdP
From this we immediately obtain more Maxwell relations:
V= (aG) ap
Still two more Maxwell relations may be obtained by considering the differential changes of the enthalpy:
H= U+ PV dH= TdS+ VdP
from which follow
(-aH) s ap
(1.36) (1.37)
Further Maxwell relations are
T- and
(au) as
p = _(
au) s av
which follow from the first law, dU = - P dV + T dS. The eight Maxwell relations may be conveniently summarized by the following diagram:
The functions A, G, H, U are flanked by their respective natural arguments, for example, A = A(V, T). The derivative with respect to one argument, with the other held fixed, may be found by going along a diagonal line, either with or against the direction of the arrow. Going against the arrow yields a minus sign; for example, (aAjaT)v= -S, (aAjavh= -Po
The second law of thermodynamics enables us to define the entropy of a substance up to an arbitrary additive constant. The definition of the entropy depends on the existence of a reversible transformation connecting an arbitrarily chosen reference state 0 to the state A under consideration. Such a reversible transformation always exists if both 0 and A lie on one sheet of the equation of state surface. If we consider two different substances, or metastable phases of the same substance, the equation of state surface may consist of more than one disjoint sheets. In such cases the kind of reversible path we have mentioned may not exist. Therefore the second law does not uniquely determine the difference in entropy of two states A and B, if A refers to one substance and B to another. In 1905 N ernst supplied a rule for such a determination. This rule has since been called the third law of thermodynamics. It states: The entropy of a system at absolute zero is a universal constant, which may be taken to be zero.
The generality of this statement rests in the facts that (a) it refers to any system, and that (b) it states that S = 0 at T = 0, regardless of the values of any other parameter of which S may be a function. It is obvious that the third law renders the entropy of any state of any system unique. The third law immediately implies that any heat capacity of a system must vanish at absolute zero. Let R be any reversible path connecting a state of the system at absolute zero to the state A, whose entropy is to be calculated. Let CR(T) be the heat capacity of the system along the path R. Then, by the second law, (1.40) But according to the third law (1.41) Hence we must have (1.42) In particular, C may be C or Cpo The statement (1.42) is experimentally R v verified for all substances so far examined. A less obvious consequence of the third law is that at absolute zero the coefficient of thermal expansion of any substance vanishes. This may be shown as follows. From the T dS equations we can deduce the equalities