5.6 The Clausius-Clapeyron equation

 

Consider two phases, 1 and 2, that are in equilibrium at pressure p and temperature T. It is known that phases in equilibrium satisfy the condition given by Eq. (5.94):

 

                                                        

 

Change the temperature of each of the phases by dT, and the pressure by dp. If in changing the temperature we change the pressure in such a way that the two phases continue to be in equilibrium at the new temperature T + dT, then, in the new conditions, the potentials of the two phases will, obviously, be equal:

 

                                                                             (5.97)

 

It is clear that the function φ (p + dp, T + dT) can be expanded in a Taylor's series:

 

                                                           (5.98)

 

In accordance with Eqs. (5.34) and (5.35), we have:

 

                                                                                                                           (5.99)

 

                                                                                                                      (5.100)

 

and, consequently, the series (5.98) can be presented in the following form:

 

                                                                             (5.101)

 

Substituting this expression in the left- and right-hand sides of Eq. (5.97), we get:

 

                                  

 

Taking into account that φ1 (p, T) = φ2 (p, T), we obtain the equation of the phase transition boundary, or the Clausius-Clapeyron equation:

 

                                                                                                                     (5.102)

 

This equation relates unambiguously the slope of the phase transition boundary on the p-T diagram with the difference between the entropies of the coexisting phases (s1 and s2) and the difference between the specific volumes of these phases (v1 and v2). Since, as we know, at a constant temperature a substance undergoes a change of phase at equal phase pressures, i.e. the process proceeds along an isobar with the temperatures of the two phases being equal, then, applied to the given case of p = const, the mathematical formulation of the second law of thermodynamics,

 

                                                                                                                     (4.29)

 

takes the following form:

 

                                                                                                                            (5.103)

 

Integrating equation (5.103) between the limits represented by arbitrary points 1 and 2 and taking into account that temperature T remains constant during the change of phase, we obtain:

 

                                                                                                           (5.104)

 

As applied to a phase transition process, the quantity (h2 – h1) is the difference between the enthalpies of the coexisting phases and represents the heat of phase transition, i.e. the amount of heat absorbed (or released) by a certain quantity of substance (usually 1 kg) in the process of change of phase.

Denoting the heat of phase change by

 

                                                                                                                         (5.105)

 

we obtain Eq. (5.104):

 

                                                                                                                         (5.106)

 

Substituting the value of the difference in entropies in Eq. (5.102), the Clausius-Clapeyron equation becomes

 

                                                                                                                 (5.107)

 

The liquid-vapor phase change

 

As applied to the liquid-vapor phase change, Eq. (5.107) takes the form

 

                                                                                                                  (5.108)

 

where r is the heat of vaporization (latent heat), v" the specific volume of vapor on the saturation line, v' the specific volume of liquid on the saturation line, and ps is the equilibrium pressure at the points of the liquid-vapor change of phase.[1]

Since the density of vapor is always less than the density of liquid, v" > v', from Eq. (5.108) it follows that for the liquid-vapor phase change the ratio dps/dT is always greater than zero, i.e. saturation pressure increases with rising temperature. The Clausius-Clapeyron equation relates the derivative of saturation pressure with respect to temperature to the caloric (r) and thermal (v' and v") properties of substance on the saturation line. Of great interest are the attempts to solve the Clausius-Clapeyron equation and to find the explicit expression for saturation pressure in relation to temperature.

If vapor pressure is low, the value of the specific volume of vapor is several orders of magnitudes greater than the value of the specific volume of liquid, v" >> v', and the difference (v" - v') in the denominator of the right-hand side of Eq. (5.108) can, therefore, be replaced by v" with a sufficient degree of accuracy. In addition, at low vapor pressures vapor can be regarded as a good approximation to an ideal gas, for which, in accordance with Clapeyron's equation,

 

                                                                                                                             (5.109)

 

Substituting the above expression into Eq. (5.108), and carrying out simple transformations, we obtain:

 

                                                                                                                  (5.110)

 

whence

 

                                                                                                   (5.111)

 

Since at low temperatures the heat of vaporization is a slowly varying function of temperature[2], the value of r can be considered to be practically independent of temperature in a number of cases. The value of r can then be taken out of the integral sign in Eq. (5.111), and we get:

 

                                                                                                         (5.112)

 

To determine the magnitude of the constant of Eq. (5.112), the saturation pressure ps must be known at some fixed temperature. The integration constant is usually determined from the standard boiling point (or temperature) Tst.b., i.e. on the boiling point of the liquid at standard atmospheric pressure, ps = 760 mm Hg = 101.325 kPa, the so-called standard atmosphere. From Eq. (5.112) we get:

 

                                                                                 (5.113)

 

Equation (5.113) is sometimes encountered in the following form:

 

                                                                                                         (5.114)

 

In dealing with formula (5.114), it should be borne in mind that the pressure ps is expressed here in standard atmospheres; in other words, the value under the logarithm sign is a dimensionless quantity, the ratio of the pressure at the sought point to the standard atmosphere. The values of ps substituted into Eq. (5.114) cannot obviously be expressed in units other than standard atmosphere, for instance, in units of kPa, kgf/cm2, etc., since the value Tst.b. in the right-hand side of the equation represents the boiling point of liquid at a pressure equal to one standard atmosphere. But if Tst.b. is replaced by the boiling point at a pressure of, for instance, 100 kPa, 1 kgf/cm2, or 1 mm Hg, then ps in the left-hand side of this equation must also be expressed in kPa, kgf/cm2, mm of Hg, etc.

To put differently, when the integration constant is determined not from the value of Tst.b. but from the known saturation pressure at some other temperature (denoted T*), Eq. (5.112) takes the form:

 

                                                                                                   (5.115)

 

From Eq. (5.112) it follows that at low pressures the dependence ln ps = f (1/T) must be linear. The treatment of experimental data on the dependence ps versus T for a great number of substances shows the dependence of ln ps on 1/T to be, in fact, linear, as illustrated in Fig. 5.5.

 

 

Fig. 5.5

 

One more point of interest must be mentioned. As was noted above, Eq. (5.112) is valid only at low pressures. In this connection it would appear that the linear nature of the dependence ln ps = f (1/T) holds only for the low pressure region. However, the treatment of experimental data shows that for many substances this linear dependence is also valid in the high pressure region, where vapor cannot be regarded as an ideal gas, the magnitude of v' should not be ignored when compared with the magnitude of v", and where r depends substantially on temperature. This is often taken as a surprise: Is this a new thermodynamic dependence or not? This can be elucidated in the following manner. Multiplying the numerator and denominator of the right-hand side of Eq. (5.108) by p, we obtain:

 

                                                           

 

and, denoting p (v" - v') by β we obtain:

 

                                                                                                                      (5.116)

 

Since, in accordance with Eq. (5.105),

 

                                                                                                                             (5.117)

 

and [see Eq. (2.47a)] we have:

 

                                                                    

 

                                                                                                         (5.118)

 

i.e. the heat of vaporization, or latent heat, can be regarded as consisting of two parts: the difference between the internal energies of the phases involved, (u" - u'), and the quantity p (v" - v'). The quantity (u" - u'), sometimes called the heat of disintegration, represents the amount of heat which must be added to a certain quantity of substance so as to overcome the intermolecular forces in the process of vaporization. The quantity β = p (v" - v') is obviously the work expended to ensure expansion of the substance from the state with specific volume v' to the state with specific volume v".[3] Experimental data have shown that r/β can be described with a sufficient degree of accuracy by the empirical formula

 

                                                                                                                            (5.119)

 

where r0 is the heat of vaporization at a temperature, and C is constant.

Substituting (5.119) into (5.116), we obtain:

 

                                                                                                                   (5.120)

 

The structure of Eq. (5.120) makes it similar to Eq. (5.110); the solution of this equation leads to the linear dependence of ln ps on 1/T.

In solving Eq. (5.110) it was assumed that since at low pressures r is a slowly varying function of temperature, it may be considered constant. For better agreement with experimental data, however, it often proves to be of an advantage to take into account the dependence of the heat of vaporization on temperature, using the simplest linear dependence

 

                                                                                                                         (5.121)

 

where the constants a and b are calculated from the experimental data on the dependence of the heat of vaporization on temperature.

Substituting (5.121) in Eq. (5.111) and solving the latter, we obtain:

 

                                                                                       (5.122)

 

whence

 

                                                                                 (5.123)

 

where ps (T*) is the known saturation pressure at the temperature T*.

The solid-liquid phase change

 

When applied to a solid-liquid phase change (melting), Eq. (5.107) can be presented as

 

                                                                                                                 (5.124)

 

where λ is the heat of fusion or heat of melting, v1 the specific volume of the liquid phase on the melting line, and vs is the specific volume of the solid phase on the melting line.

Since the densities of a substance in the liquid and solid phases are quantities of the same order of magnitude, v1 and vs are usually close to each other, and two cases are possible: (1) the density of the solid is greater than that of the liquid (v1 > vs), and (2) the density of the solid is smaller than that of the liquid (v1 < vs). It is clear from Eq. (5.124) that for the first case dp/dT > 0, i.e. the fusion point rises with rising pressure. For the second case the quantity dp/dT, as can be seen from Eq. (5.124), happens to be negative, i.e. the fusion point lowers with rising pressure. Thus, for some substances (water, for instance) the slope of the melting boundary on the p-T diagram is negative (see Fig. 5.3).

 

The solid-vapor phase change

 

When applied to a solid-vapor phase change, Eq. (5.107) takes the following form:

 

                                                                                                                (5.125)

 

where L is the heat of sublimation, vv the specific volume of vapor on the sublimation line, and vs is the specific volume of the solid.

Since sublimation usually takes place at low pressures (below the pressure at the triple point), the specific volume of the forming vapor on the sublimation line is several orders of magnitude greater than the specific volume of the solid phase. Therefore, in accordance with Eq. (5.125), for the solid-vapor phase change the quantity dp/dT is greater than zero, which means that the slope of the sublimation line in the p-T diagram is positive.

It is clear that in the process of sublimation vy > vs and the value vs in Eq. (5.125) can be ignored. The specific volume of vapor on the sublimation line is described very accurately by Clapeyron's equation

 

                                                                

 

and Eq. (5.125) takes the form

 

                                                                                                                   (5.126)

 

Knowing the temperature dependence for the heat of sublimation, we can solve Eq. (5.126).

In conclusion, one more interesting application of the Clausius-Clapeyron equation will be indicated. As was already mentioned in Sec. 3.4, of great importance is the introduction of corrections to any empirical (practical) temperature scale so as to reduce this scale to the thermodynamic temperature scale, i.e. to establish a thermodynamic temperature scale based on a given concrete empirical temperature scale (for instance, on that of a gas thermometer). The equation for the corrections to the International Temperature Scale, used to convert it into the thermodynamic scale, was given in Chapter 3. But how can these corrections be determined? These corrections, i.e. the difference between the temperatures on the thermodynamic, T, and empirical, T*, scales, or, in other words, the dependence T = f (T*), can be determined by various methods, one of which is based on the Clausius-Clapeyron equation.

As was already mentioned repeatedly, the thermodynamic temperature is present in all thermodynamic relationships, including the Clausius-Clapeyron equation (5.108), which, when applied to the liquid-vapor phase change, takes the following form:

 

                                                                                                                 (5.108a)

 

If the values of v', v" and r are determined experimentally, Eq. (5.108a) can obviously be used to calculate dT/dp.

On the other hand, if we have experimental data on the dependence of the saturation pressure ps on temperature (where temperature measurements are taken with the aid of a thermometer graduated according to the empirical scale, i.e. T* is measured), the magnitude of the derivative dT*/dp can be calculated by means of graphical or numerical methods.

The value of dT/dp, calculated from Eq. (5.108a), is evidently related with dT*/dp via the relationship

 

                                                          

 

Substituting the value of dT/dp from relation (5.108a), we obtain:

 

                                                      

 

Integrating this relationship, we have:

 

                                                    

 

whence

 

                                                

 

Thus, if some thermodynamic-scale temperature T1 corresponding to the temperature T*, is known, then, having experimental data on v', v", r and p in the temperature interval from  to  the derived equation can be used to calculate the thermodynamic temperature T2, corresponding to the temperature . If T1 is assumed to be equal to  at some point (the only fixed point), this equation can be used to find the correction for any temperature on the empirical temperature scale.

 

 

 


[1] The index s will always refer to values on the saturation line and the indexes prime and double prime, respectively, to the liquid and vapor in the state of saturation.

[2] For greater detail on the temperature dependence of r, see Sec. 6.6.

[3] The independent consideration of the quantities (u" - u') and p (v" - v') is, strictly speaking, conventional to a certain degree, since the breakdown of associations is intimately associated with an increase in the specific volume of a substance.