*The Clapeyron Equation*

In a one-component phase diagram the lines separating the phases (for
example, solid from liquid, liquid from vapor, and so on) are not arbitrary,
they are determined by the principles of thermodynamics and equilibrium.
Thermodynamics provides an equation for each line which gives pressure,
*p*, as a function of temperature *T,* in other words an equation
of the form *p = p(T)* . It is now our task to find the form of the
function *p(T)*. The diagram below is a portion of a hypothetical
phase diagram and the phase line on this diagram separates the phases α
and β. In other words, the phases α
and β are in equilibrium with each other at
any and all points on the line.

Phase transitions occur at constant temperature and pressure at the
points on this line. Consider the phase transition,

α → β. (1)Since the α and β phases are in equilibrium with each other at any point on the line we know that the change in the Gibbs free energy for the transition given by Equation 1 is zero everywhere on the α-β phase line. That is,

ΔWe already know that for any process at constant temperature it must be true thatG= 0. (2)

Δso thatG= ΔH−TΔS, (3)

(4)for a phase transition.

The various quantities appearing in these equations are defined as usual by,

ΔWe will also define ΔG = G, (5)_{β}− G_{α}Δ

S = S. (6)_{β}− S_{α}Δ

H=H_{β}−H_{α}. (7)

ΔAs we have already said, ΔV = V. (8)_{β}− V_{α}

. (9)We know from

(10)and

. (11)Using Equations10 and 11 in Equation 9 (the two minus signs cancel) we get,

. (12)Equation 12 is the Clapeyron equation. The Clapeyron equation is thermodynamically exact. It contains no approximations. There is another version of the Clapeyron equation which we obtain by inserting Equation 4 for Δ

. (13)Equation 13 is useful when we want to integrate

. (14)To integrate Equation 14 we must know how Δ

. (15)If we let

(16)

*The Clausius-Clapeyron Equation*

Equations 12 and 13 are exact and are valid for all types of phase transitions,
solid → liquid, solid →
gas, liquid → gas, and so on. However, for
solid → gas and liquid →
gas phase transitions we can make two approximations which will give us
a very useful equation called the Calusius-Clapeyron equation. The first
approximation comes from noting that the volume of a given amount of gas
is much larger that the volume of an equivalent amount of solid or liquid.
This being true, we can approximate
Δ*V*
by the volume of the gas. That is,

ΔorV=V_{gas}−V_{liquid}≈V_{gas}(17)

ΔIn the second approximation we replaceV=V_{gas}−V_{solid}≈V_{gas}. (18)

, (19)where the bar over the enthalpy,

. (20)Equation 20 can be integrated if we know how Δ

(21)Equation 21 is the integrated form of the Clausius-Clapeyron equation. If we want pressure as a function of temperature we can let the point (

. (22)Notice that if we know two points on the curve we can solve for Δ

*Other details and interesting stuff*

1. You can get fancy with Equations 19 and 20 if you want. For example,
we know that Δ*H* for a phase transition
is not really constant. If we wanted to take into account the temperature
dependence of Δ*H* we need to know the
heat capacities of the two phases, or rather the difference in the heat
capacity for the two phases,

ΔThenC=_{p}C. (23)_{pβ}− C_{pα}

. (24)Initially we regarded Δ

. (25)Using this expression for Δ

, (26)or,

. (27)Equation 27 integrates to,

(28)You can get even fancier. The next level of approximation would be to include the temperature dependence of the heat capacities, but we won't deal with that here.

2. Equation 12 does not depend on Δ*G*
being zero. It only assumes that Δ*G* is
constant for the α →
β transition. Of course, if Δ*G*
is not zero then equation 4 is not true which makes Equation 13 invalid.
Further, if Δ*G* is not zero then either
the α phase or the β
phase is thermodynamically unstable. (We usually use the word *metastable*
to describe this situation. Examples are supercooled liquids, superheated
liquids, supercooled vapors, and some metastable solids - like diamond.)
Nevertheless, one can visualize a family of curves running along side the
phase diagram curve with Δ*G* constant
at some value other than zero.

WRS

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Last updated 1 Dec 05

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