A note on integrating the Clapeyron equation without neglecting the specific fluid volume

Under certain approximations, the Clapeyron equation can be integrated to yield a simple exponential relation giving the saturation vapor pressure over a condensed phase as a function of temperature. The derivation usually assumes that the vapor behaves as an ideal gas with constant specific heat, and that the fluid also has constant specific heat. In addition, the specific fluid volume is neglected in comparison with the specific vapor volume. In this case, the Clapeyron equation is separable and readily integrable in closed form. It is shown here that this latter assumption is not required. A simple closed‐form relation between saturation vapor pressure and temperature is derived which includes the condensed phase‐specific volume. Two examples of the use of this result are presented.


| INTRODUCTION
The Clapeyron equation, is often evoked when considering the temperature dependence of the pressure of a vapor phase in equilibrium with its condensed phase.The nomenclature in Equation ( 1), and used throughout this paper, is gathered in Table A1.
If the vapor is modeled as a perfect gas, whose specific volume v g is considered so much larger than the condensed phase-specific volume v f that the later can be ignored, and the heat of transition is assumed to vary linearly with temperature, then this equation can be integrated in closed form yielding well-known expressions for the saturation pressure as a function of temperature.
In this note, it is shown that it is not necessary to ignore the condensed volume.A closed-form relation between the saturation pressure and temperature is derived below that includes the condensed phase-specific volume.The resulting expression proves useful in analyzing systems involving liquid-vapor transitions.Some examples are presented in Sections 5 and 6.
The following assumptions regarding state functions are assumed throughout this paper: • The vapor phase behaves as a perfect gas with constant-specific heat.• The condensed phase also has constant-specific heat and has zero compressibility and thermal expansion coefficient.
We assume the substance of interest is pure water and explore the liquid-to-vapor-phase properties; however, the treatment is applicable to any condensed-to-vapor transition at temperatures well below the critical temperature.
The assumed state functions follow standard practice, such as covered in Wark (1995).

| AN ACCURATE EXPRESSION FOR THE HEAT OF VAPORIZATION
To integrate the Clapeyron equation, we need an expression for the temperature dependence of the heat of vaporization, h fg T ð Þ.The expression usually encountered for this is has a linear dependence in temperature, T.
where T R is a reference temperature, and c f and c p,v are the constant pressure-specific heats of the fluid and vapor.See for example (Emanuel, 1994, p. 115), (Ambaum, 2020), or (Romps, 2021).In this section, we derive a sometimes-used alternative to Equation (2).

| Enthalpy
The specific enthalpy of liquid (fluid) water, h f , is where u f is the specific internal energy, v f the specific volume, and p the pressure.We will need the temperature and pressure dependence of h f .In principle for this simple compressible substance all the terms in the above equation are functions of T and p.In this work, we need the enthalpy only along the liquid-vapor saturation line where p ¼ p sat T ð Þ, and so u f , v f , and h f depend only on T. In other words If we assume that the internal energy at the saturation pressure is linearly dependent on temperature change for small changes, and that the specific volume is independent of temperature and pressure, the saturation enthalpy of the fluid is then where c f and v f are constants.This raises the question of what commonly defined specific heat best approximates c f .Since the fluid is incompressible, its internal energy is independent of pressure.For small changes in temperature and pressure So, except for very small factors related to the compressibility of liquid water, c f in Equation ( 4) is indistinguishable from constant pressure-specific heat of the fluid, c p:f .For the purposes here it is merely required that u f vary linearly with temperature along the saturation line.c f for water can, of course be found from the saturated steam tables (Wagner & Pruss, 2002).It differs negligibly from c p,f , but substantially from c p,v .
The enthalpy of the saturated vapor is Since the enthalpy of an ideal gas depends only on temperature, Equation ( 5) is valid at pressures other than saturation.For the case of water, Equations ( 4) and ( 5) for the enthalpy of liquid and vapor water are often referred to in the HVAC literature as the Wepfer approximations (Wepfer et al., 1979).h fg is temperature dependent.This dependence can be found by comparing two paths to h v .The first is that of Equation (4); heat the fluid from T R to T, then increment by p sat T ð Þv f to get the enthalpy, and finally vaporize at T. The other path is to first vaporize the substance at T R and then heat the vapor to T. This gives where c p,v is the heat capacity of the vapor at constant pressure.
Comparing the two above equations yields A form which will be useful is to write Equation (6) as where L o is a constant often called the "extrapolated heat of vaporization."Again, L o is normally defined without the p sat T ð Þv f term (Emanuel, 1994, p. 116).The above discussion was presented for the purpose of introducing the 6), ( 7), and ( 8), which will be needed.
In Equations ( 3) to (8) the term p sat v f is usually small and can be, and usually is, neglected.The terms in v f can appear at many points in an analysis, however, and if not included at each point, inconsistent results are obtained.This showed up in the authors' analysis of the minimum work required to condense water vapor from air, where neglecting the v f terms resulted in an expression differing form the known work for a reversible process (Swanson, 2023).That result was the impetus for this work.

| Entropy
It will prove useful to also have the specific entropy.The specific entropy of liquid water, still assumed incompressible and setting the reference entropy to zero at and for water vapor The last term is from the standard relation s g À s f ¼ h fg =T, while the second accounts for changes in pressure from p sat .

| INTEGRATING THE CLAPEYRON EQUATION
The vapor pressure as a function of temperature can be found by integrating the Clapeyron equation (Equation ( 1)).Using Equation ( 7) for h fg , and assuming the vapor is a perfect gas, that is,

| Approximate form neglecting fluid volume
Equation ( 11) is not separable as written, and thus, it is not obvious how to perform the integration in closed form.The terms involving v f are quite small, however, and thus it is customary to neglect them.For example, for water at 25 C, v f =v g ¼ p sat T ð Þv f =R v T ¼ 2:3Â 10 À5 .After neglecting the terms in v f , Equation ( 7) becomes which is separable and readily integrable.Integrating from the reference temperature, T R , to a temperature T gives the standard textbook relation Equation (13a) (Emanuel, 1994, p. 116), (Romps, 2021) ln or 8)) in our present approximation of setting v f ¼ 0, we can also write Equation (13a) as To the author's knowledge, the form of Equation (13c) was first elucidated by (Ambaum, 2020).

| Integration including the fluid volume
Fortunately, the full Clapeyron equation (Equation ( 11)) can be separated by means of a change in variable rendering it unnecessary to neglect v f .If we let the dimensionless variable u, where u is a function only of T, be given by then it can be shown that Inserting this into Equation ( 11) gives the separated form Inserting Equation ( 14) into this gives This is the result sought.Equation (15a) has the same form as (13a), but with the additional terms in p sat v f .
Noting that since It seems quite interesting that when one includes the v f terms the solution of the Clapeyron equation, as in Equation (15b), it has the identical form of (13c) where v f ¼ 0 had been assumed.The impact of this term is implicitly included in h fg T ð Þthrough Equation ( 6).It should be noted that, while Equations (15a,b) give an explicit relation between T and p sat T ð Þ, it is not possible to solve for p sat T ð Þ in terms of T as it is in the case where the v f terms are ignored (Equation (13b)).
That Equation (15b) satisfies the Clapeyron equation can be shown by direct substitution.Differentiating (15b) with respect to temperature gives which is just the Clapeyron equation (Equation ( 11)).

| AN ALTERNATIVE DERIVATION OF VAPOR PRESSURE VERSUS TEMPERATURE
As pointed out by Ambaum (Ambaum, 2020), when the v f terms are ignored the above state equations (Equations ( 6), (9), and (10)) are sufficient to derive an expression for the vapor pressure as a function of temperature without having to integrate the Clapeyron equation (Equation ( 11)).This is done by performing the same maneuver with entropy that was done with enthalpy to determine the temperature dependence of h fg in deriving Equation (6).We now show that the same obtains when the v f terms are included.
We can think of two routes to the vapor entropy.From Equation (8) the saturated vapor entropy at temperature T is We can also start at the saturated vapor entropy at Treating the vapor as an ideal gas we can go from this point to a different temperature and pressure using the ideal gas relation connecting points of different temperature and pressure.
ð Þ for the saturation entropy this becomes Equating Equations ( 16) and ( 17) gives which is the same as Equation (15b).

| AN EXAMPLE OF THE NEED FOR INCLUDING THE LIQUID VOLUME IN ENTHALPY
We have seen that incorporating the v f term in liquid and vapor enthalpy (Equations ( 3) and ( 4)) allows the vapor pressure to satisfy the Clapeyron equation exactly.This need becomes apparent in many analyses.As a simple example, suppose we have a piston and cylinder arrangement containing pure water vapor of mass M w and volume V v at the saturation pressure.We then compress it isothermally and reversibly to pure liquid with volume V f .The work in doing this (neglecting the force from the ambient pressure) is clearly We can compute this work another way by realizing that for this isothermal process which exchanges heat with the ambient at temperature T, the work is also the change in exergy, ΔE, of the system (Wark, 1995, p. 80), where E ¼ E À TS (again neglecting atmospheric work).Since E ¼ H À pV this becomes in agreement with the first approach.If the v f term in enthalpy were neglected, then we would have obtained instead which is clearly incorrect.The author found numerous other instances of this requirement while looking at various idealized atmospheric water harvesting cycles (Swanson, 2023).

| IMPACT OF THE NEW VAPOR PRESSURE FORMULA ON THE RELATION OF RELATIVE HUMIDITY TO DEW POINT
The above equations for vapor pressure can be used to determine the relation between relative humidity and dew point temperature (Romps, 2021).The usual procedure is to define the atmosphere's relative humidity, H, as , where p v is the actual water partial pressure and p sat T a ð Þ is the vapor pressure under saturation conditions at the air temperature, T a .If we lower the temperature at constant total pressure to the point that the vapor is saturated, this temperature is the dewpoint, where p v ¼ p sat T dp À Á .Note that p v remains constant at constant total pressure assuming ideal gasses.Thus, Let us first neglect the impact of the v f terms.Inserting this into Equation (13a) gives It might be imagined that the impact of non-zero v f can be found by inserting Equation ( 19) for H into Equation (15a) which gives We now show that this is incorrect.The terms in p sat v f should not be there.The problem comes from the fact that we have used equations for vapor pressure for the case of pure water vapor over pure water.The presence of non-condensable gasses (nitrogen, oxygen, etc.) exerts an additional pressure on the liquid so that the saturation pressure in the atmosphere is larger than over pure water.This is called the Poynting effect (Wark, 1995, p. 344).If p sat is the water saturation pressure of pure vapor over water, then the effect of the remaining atmospheric constituents is to raise the saturation pressure in the atmosphere, denoted by p a sat , by were p atm is the total atmospheric pressure.This effect can be thought of as arising from the increase in pressure beyond p sat T ð Þ on the liquid raising its enthalpy and hence its chemical potential.That necessitates a similar increase in the chemical potential of the vapor.
In the presence of the atmospheric gasses besides water vapor the actual relative humidity becomes where L 0 0 ¼ L 0 À p atm v f .The p sat v f terms cancel so this has the same form as Equation ( 20), but with the change in extrapolated heat of vaporization from L 0 to L 0 À p atm v f .For the case of water around room temperature and atmospheric pressure, p atm v f ¼ 100 kJ=kg, whereas L 0 ¼ 3.14 Â 106 kJ/kg.So the change is quite negligible.
This result agrees with Ambaum who argues that because the chemical potential of the liquid and vapor water must be equal even in the presence of the remaining atmospheric gasses, and because the liquid volume is the derivative of the Gibbs function with respect to pressure, the specific liquid volume can be ignored at constant pressure (Ambaum, 2020).
It should be noted that this result obtains only when the Poynting effect is used to correct the relative humidity, and the specific fluid volume is retained in the Clapeyron equation.Having only one or the other correction will give an erroneous result.

| DISCUSSION
The Clapeyron equation has been integrated to obtain a simple expression for vapor pressure as a function of temperature that does not neglect the condensed vapor volume.This expression can be used in thermodynamic cycle analyses to yield results that strictly obey the First and Second Laws.It seems surprising that any new result can be found in an equation that dates back almost 200 years (Wisniak, 2001).More recently, however, Shilo and Ghez did just that (Shilo & Ghez, 2008).They transformed the Clapeyron equation into a Bernoulli equation with known solutions.Their method uses a transformation with the dimensionless variable p sat v f =h fg (in the notation of this paper), whereas we use p sat v f =R v T and obtain an explicit solution that involves our approximation for h fg (Equation ( 6)).The temperature dependence of h fg used in this paper, which contains terms in both T and p sat T ð Þ, is not apparent in their formulation.It should not be expected that Equation (15a) is a particularly more accurate representation of experimental results than Equation (13a) which neglects the impact of v f , because this term is so small.The impact of other approximations such as a perfect gas for the vapor (i.e., unity compressibility factor) and constant specific heats inject more serious errors.For example, using the data of Wagner and Pruss (Wagner & Pruss, 2002) the compressibility factor of saturated water vapor at 25 C is 0.9985, for a relative error of 1.5 Â 10 À3 , and the specific heat of liquid water varies from 4217 kJ/kg to 4182 kJ/kg over the range 0 to 25 C, for a relative difference of 8.4 Â 10 À3 .On the other hand, it can be shown that the vapor pressure for water calculated at 25 C using the exact result (Equation (15a)) differs from the standard result (Equation (13a)) by only a relative difference of 7 Â 10 À6 , which is quite less than the above differences.For this calculation, we used c f = 4195, c p,v = 1850, and h fg 0 C ð Þ = 2.501 Â 10 6 J/kg.
Both tabulated and parameter fit representations of the experimental thermodynamic values obviously avoid all the approximations inherent in this simple model and would be preferred when high accuracy is needed.For example, very high accuracy for saturation pressure has been obtained by using polynomial expansions in T for the heat of vaporization, which retain accuracy close to the critical temperature where the ideal gas law is no longer applicable (Leibowici & Nichita, 2014).The main impact of the new vapor pressure formula is to render idealized cycle analyses to be in strict compliance with the First and Second Laws.Its appeal is more esthetic than practical in this sense.