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Fig. 2.1 Surface of equation of state of a typical substance (not to scale).
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P(T), the Gibbs potential of this state must be at a minimum. That is, if any parameters other than T and P are varied slightly, we must have 8G = O. Let us
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vary the composition of the mixture by converting an amount 8m of liquid to gas, so that
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The total Gibbs potential of the gas-liquid mixture may be represented, with neglect of surface effects, as
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where gl is the Gibbs potential per unit mass of the liquid in state 1 and g2 is that for the gas in state 2. They are also called chemical potentials. They are independent of the total mass of the phases but may depend on the density of the phases (which, however, are not altered when we transfer mass from one phase to the other). Thus
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Flg.2.2 P- V and P- T diagrams of a typical substance (not to scale).
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Fig. 2.3 An isotherm exhibiting a phase transition.
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The condition for equilibrium is then gi = gz (2.3) This condition determines the vapor pressure, as we shall see. The chemical potentials gl(P, T) and gz(P, T) are two state functions of the liquid and gas respectively. Recall that in each phase we have (entropy per unit mass) (volume per unit mass)
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(2.4) (2.5)
All liquid
Gasliquid mixture
liquid mixture
Fig. 2.4 Schematic illustration of a first-order phase transition. The temperature and the pressure of the system remain constant throughout the transition. The total volume of the system changes as the relative amount of the substance in the two phases changes, because the two phases have different densities.
(T constant)
+Transition tern perature
Vapor pressure
Fig. 2.5 Chemical potentials gl' g2 for the two phases m a
first-order phase transition. We see that the first derivative of gl is different from that of g2 at the transition temperature and pressure:
a(g2 - gl) aT
-(S2- S1)<O
V2 -
This is why the transition is called "first-order." The behavior of gl(P, T) and g2(P, T) are qualitatively sketched in Fig. 2.5. To determine the vapor pressure we proceed as follows. Let
= g2 - gl
I1s = S2 - SI
where all quantities are evaluated at the transition temperature T and vapor pressure P. The condition for equilibrium is that T and P be such as to make I1g = O. Dividing (2.6) by (2.7), we have
(a I1g/aT)p (a I1g/aph
By the chain relation,
( al1g) p (aT) !J.g (ap) T aT ap a I1g
(a I1g/aT)p (a I1g/ aph
The reason the chain relation is valid here is that I1g is a function of T and P,
and hence there must exist a relation of the form f(T, P, dg) =
dP(T) ( -- - dT aT ~g=O
o. The derivative
is precisely the derivative of the vapor pressure with respect to temperature under equilibrium conditions, for dg is held fixed at the value zero. Combining (2.11), (2.10), and (2.9), we obtain dP(T) ds ---=(2.12) dT dv The quantity (2.13) 1= Tds is called the latent heat of transition. Thus
(2.14) dT Tdv This is known as the Clapeyron equation. It governs the vapor pressure in any first-order transition. It may happen in a phase transition that S2 - SI = 0 and v2 - VI = O. When this is so the first derivatives of the chemical potentials are continuous across the transition point. Such a transition is not of the first order and would not be governed by the Clapeyron equation, and its isotherm would not have a horizontal part in the P- V diagram. Ehrenfest defines a phase transition to be an nth-order transition if, at the transition point,
1 2 --*-n n aT aT
whereas all lower derivatives are equal. A well-known example is the second-order transition in superconductivity. On the other hand many examples of phase transitions cannot be described by this scheme. Notable among these are the Curie point transition in ferromagnets, the order-disorder transition in binary alloys, and the .\ transition in liquid helium. In these cases the specific heat diverges logarithmically at the transition point. Since the specific heat is related to the second derivative of g these examples cannot be characterized by the behaviors of the higher derivatives of g, because they do not exist. Modem usuage distinguishes only between first-order and higher-order transitions, and the latter are usually indiscriminately called "second-order" transitions.