The Clapeyron equation

The condition for equilibrium between two phases, a and of a pure substance is 

If the analytical forms of the functions and fip were known, it would be possible, in principle at least, to solve Eq. (12.5) for 

Equation (12.6a) expresses the fact, illustrated in Fig. 12.3(b), that the equilibrium temperature depends on the pressure. 

In the absence of this detailed knowledge of the functions ix and jXp, it is possible nonetheless to obtain a value for the derivative of the temperature with respect to pressure. Consider the equilibrium between two phases a and P under a pressure p the equilibrium temperature is T. Then, at T and p, we have 

If the pressure is changed to a value p -F dp, the equilibrium temperature will change to T + dT, and the value of each p will change to p + dp. Hence at T dT, p dp the equilibrium condition is 

The previous discussion detailed general trends in the behavior of equilibria. In order to get more quantitative, we need to derive some new expressions. 

Equation 6.3, when generalized, states that the chemical potential of two phases of the same component are equal at equilibrium  

By analogy to the natural variable expression for G, at a constant total amount of substance the infinitesimal change in fx, d/x, as pressure and temperature change infinitesimally is given by the equation 

Because the temperature change dT and pressure change dp are experienced by both phases simultaneously, there is no need to put labels on them. However, each phase will have its own characteristic molar entropy and molar volume, so each S and V must have a label to distinguish it. We can rearrange to collect the dp terms and the dT terms on opposite sides  

Unless otherwise noted, all art on this page is Cengage Learning 2014. 

FIGURE 5.1 Simplified schematic of the device for measuriDg the vapor pressure. 

If two phases (1 and 2) of a pure substance are in equilibrium (gas-liquid, gas-solid, liquid-liquid, liquid-solid, or solid-solid) at Tj and Pi, then the Gibbs energy per mol (or per Ibm or kg) must be the same in each of the equilibrium phases. 

FIGURE 5.2 Vapor pressure of 27 compounds as a function of temperature. (From Brown, G. G. etal. Unit Operations. 1951, Wiley New York. Reprintedby permission of John Wiley and Sons.) 

If we start with two coexisting phases, a and P, of a pure substance and change the temperature of both phases equally without changing the pressure, the phases will no longer be 

Equation 8.4.4 is one form of the Clapeyron equation, which contains no approximations We find an alternative form by substituting AtrsiS = Atrs-f /Tlrs (Eq. 8.3.5)  

Most substances expand on melting, making the slope of the solid-liquid coexistence curve positive. This is trae of carbon dioxide, although in Fig. 8.2(c) on page 201 the curve is so steep that it is difQcult to see the slope is positive. Exceptions at ordinary pressures, substances that contract on melting, are H2O, rubidium nitrate, and the elements antimony, bismuth, and gaUium. 

The phase diagram for H2O in Fig. 8.4 on page 203 clearly shows that the coexistence curve for ice I and liquid has a negative slope due to ordinary ice being less dense than Uquid water. The high-pressure forms of ice are more dense than the liquid, causing the slopes of the other solid-liquid coexistence curves to be positive. The ice Vll-ice VIII coexistence curve is vertical, because these two forms of ice have identical crystal structures, except for the orientations of the H2O molecule therefore, within experimental uncertainty, the two forms have equal molar volumes. 

We may rearrange Eq. 8.4.5 to give the variation of p with T along the coexistence curve  

In this section, we derive the Clapeyron equation. This equation relates changes in the pressure to changes in the temperature along a two-phase coexistence curve (e.g., the vapor pressure curve or the melting curve). Note that the condition for equilibrium between two phases is given by 

This is one form of the Clapeyron equation. It relates the slope of the coexistence curve to the entropy change and volume change of the phase transition. 

Entropy is not directly measureable, and, therefore, the Clapeyron equation as written above is not in a convenient form. However, we can relate entropy changes to enthalpy changes, which can be directly measured. At equilibrium, we have 

the entropy change of a phase transition, which is not directly measureable, can be determined from the enthalpy change of tlie phase transition, which is directly measureable. 

This is file more commonly used form of the Clapeyron equation. 

Entropy and molar volume can be quantified using a well-known equation describing the slope of any two-phase reaction boundary on a pressure-temperature diagram. From equation (8.2) it follows that 

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The analogue of the Clapeyron equation for multicomponent systems can be derived by a complex procedure of systematically eliminating the various chemical potentials, but an alternative derivation uses the Maxwell relation (A2.1.41)... 

The Clapeyron equation expresses the dynamic equilibrium existing between the vapor and the condensed phase of a pure substance ... 

Another well-known thermodynamic result, the Clapeyron equation, applies to first-order transitions (subscript 1) ... 

This expression describes the variation of the pressure-temperature coordinates of a first-order transition in terms of the changes in S and V which occur there. The Clapeyron equation cannot be applied to a second-order transition (subscript 2), because ASj and AVj are zero and their ratio is undefined for the second-order case. However, we may apply L Hopital s rule to both the numerator and denominator of the right-hand side of Eq. (4.47) to establish the limiting value of dp/dTj. In this procedure we may differentiate either with respect to p. 

This equation follows from equation 66, because vaporization occurs at the constant pressure Moreover, the heat of vaporization is related to the slope of the vapor—Hquid saturation curve through the Clapeyron equation ... 

Cluusius-Clupeyron Eijliation. Derived from equation 1, the Clapeyron equation is a fundamental relationship between the latent heat accompanying a phase change and pressure—volume—temperature (PVT data for the system (1) ... 

The Clapeyron equation is most often used to represent the relationship between the temperature dependence of a pure hquid s vapor pressure curve and its latent heat of vaporization. In this case, dT is the slope of the vapor pressure—temperature curve, ADis the difference between the... 

Correlation Methods Vapor pressure is correlated as a function of temperature by numerous methods mainly derived from the Clapeyron equation discussed in the section on enthalpy of vaporization. The classic simple equation used for correlation of low to moderate vapor pressures is the Antoine S equation (2-27). 

Known as the Clapeyron equation, this is an exacl thermodynamic relation, providing a vital connection between the properties of the liquid and vapor phases. Its use presupposes knowledge of a suitable vapor pressure vs. temperature relation. Empirical in nature, such relations are approximated by the equation... 

This differs from the water pressure equation of Clapeyron, which lacks the last term. If pj = 0 or dp, = 0, then Eq. (4.102) is identical to the Clapeyron equation, as it should be. 

While the Gibbs phase rule provides for a qualitative explanation, we can apply the Clapeyron equation, derived earlier [equation (5.71)], in conjunction with studying the temperature and pressure dependences of the chemical potential, to explain quantitatively some of the features of the one-component phase diagram. 

Line db in Figure 8.1 represents the equilibrium melting line for C02. Note that the equilibrium pressure is very nearly a linear function of T in the (p, T) range shown in this portion of the graph, and that the slope of the line, (d/ /d7 )s ], is positive and very steep, with a magnitude of approximately 5 MPa-K-1. These observations can be explained using the Clapeyron equation. For the process... 

The Clausius-Clapeyron equation The Clapeyron equation can be used to derive an approximate equation that relates the vapor pressure of a liquid or solid to temperature. For the vaporization process... 

The Clapeyron equation applies to these transitions so that... 

We will be looking at first-order phase transitions in a mixture so that the Clapeyron equation, as well as the Gibbs phase rule, apply. We will describe mostly binary systems so that C = 2 and the phase rule becomes... 

This equation is called the Clapeyron equation and can be applied to any two phases in equilibrium, e.g., solid and liquid, liquid and vapor, solid and vapor or two crystalline forms of the same solid. Thus for the equilibrium... 

It is of interest to consider the variation of vapor pressure with temperature. The vapor pressure of a liquid is constant at a given temperature. It increases with increasing temperature upto the critical temperature of the liquid. The liquid is completely in the vapor state above the critical temperature. The variation of the vapor pressure with temperature can be expressed mathematically by the Clapeyron-Clausius equation. Clausius modified the Clapeyron equation in the following manner by assuming that the vapor behaves like an ideal gas. 

If the temperature is not near the critical temperature, the volume of a liquid can be considered to be negligibly small compared with the volume of the vapor. The Clapeyron equation then becomes... 

Since the molar volume of a gas is much larger than the molar volume of the liquid phase, the Clapeyron equation can be reduced to... 

We are now able to use this model for the Gibbs energy of the liquid to calculate the melting line for four-coordinated Si by using the Clapeyron equation (eq. 2.10) ... 

We move from the qualitative argument that r(meit) decreases as p increases, and next look for a quantitative measure of the changes in melt temperature with pressure. We will employ the Clapeyron equation ... 

Strategy. (1) Calculate the pressure exerted and hence the pressure change. (2) Insert values into the Clapeyron equation (Equation (5.1)). 

Before inserting values into the Clapeyron equation, we rearrange it slightly, first by multiplying both sides by TAV, then dividing both sides by AH , to give... 

Inserting this relationship into Equation (5.4) yields the Clapeyron equation in its familiar form. 

In fact, Equation (5.4) is also called the Clapeyron equation. This equation holds for phase changes between any two phases and, at heart, quantitatively defines the phase... 

We are permitted to assume that dp is directly proportional to dT because AH and AV are regarded as constants, although even a casual inspection of a phase diagram shows how curved the solid-gas and liquid-gas phase boundaries are. Such curvature clearly indicates that the Clapeyron equation fails to work except over extremely limited ranges of p and T. Why ... 

The Clapeyron equation fails to work for phase changes involving gases, except over extremely limited ranges of p and T. 

The worst deviations from the Clapeyron equation occur when one of the phases is a gas. This occurs because the volume of a gas depends strongly on temperature, whereas the volume of a liquid or solid does not. Accordingly, the value of A Vm is not independent of temperature when the equilibrium involves a gas. 

We saw from the Clapeyron equation, Equation (5.1), how the decrease in freezing temperature dT is proportional to the applied pressure dp, so one of the easiest ways of avoiding the lethal conversion of solid ice forming liquid water is to apply a smaller pressure - which will decrease dT in direct proportion. 

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