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Clausius-Clapeyron equation and saturation vapour pressure:

simple theory reconciled with practice

Demetris Koutsoyiannis

Department of Water Resources and Environmental Engineering, School of Civil

Engineering, National Technical University of Athens, Greece

E-mail: dk@itia.ntua.gr; URL: http://www.itia.ntua.gr/dk

Abstract. While the Clausius-Clapeyron equation is very important as it determines

the saturation vapour pressure, in practice it is replaced by empirical, typically

Magnus type, equations which are more accurate. It is shown that the reduced

accuracy reflects an inconsistent assumption that the latent heat of vaporization is

constant. Not only is this assumption unnecessary and excessive, but it is also

contradictory to entropy maximization. Removing this assumption and using a pure

entropy maximization framework we obtain a simple closed solution, which is both

theoretically consistent and accurate. Our discussion and derivation are relevant to

students and specialists in statistical thermophysics and in geophysical sciences, and

our results are ready for practical application in physics as well as in such disciplines

as hydrology, meteorology and climatology.

Keywords: entropy, maximum entropy, saturation vapour pressure, Clausius-Clapeyron

equation, Magnus equation.

1. Introduction

The Clausius–Clapeyron relationship characterizes the transition between two phases of matter. The

importance of this equation cannot be overemphasized and transcends thermodynamics and physics.

As it determines the saturation vapour pressure for water, it provides the physical basis of the

hydrological cycle and becomes a principal equation in hydrology, meteorology, climatology, and

other geophysical sciences. Specifically, the saturation vapour pressure, also known as equilibrium

vapour pressure, is an upper limit of the quantity of vapour that the atmosphere can contain. When

this limit is reached, no additional liquid water is evaporated, while below the limit more water

evaporates. This limit is expressed in terms of the partial pressure of the vapour. At standard

temperature and pressure (STP) conditions, i.e. at a temperature of 273.15 K (0°C) the saturation

vapour pressure is 6.11 hPa, i.e. 0.611% of the total pressure of 1000 hPa. The saturation vapour

pressure increases at higher temperatures, e.g. at 25°C it is over five times higher. Conversely, when

moist air ascends and its temperature decreases, so does the saturation vapour pressure. Vapour in

excess of the lower saturation pressure starts to condense, giving rise to the formation of clouds.

The Clausius-Clapeyron equation derives from entropy maximization, which determines the

equilibrium between two phases of a substance. Mathematically it is expressed as the relationship

between temperature, T, and pressure, p, at the equilibrium. Usually it is expressed in differential form

[e.g. 1, 2; see also below]. However, in addition to the differential form, most texts [e.g. 3, p. 98; 4, p.

180; 5, p. 612; 6, p. 300] as well as electronic sources, provide an analytical equation written as

p = CT

1

−

RT

L

exp (1)

where CT

1

is an integration constant (generally, we use the symbol CT

i

to denote integration

constants), L is the latent heat of vaporization and R is the specific gas constant. This is supposed to

be the integration of the differential form under the assumption that L is constant. However, as will be

shown, this assumption is inconsistent with the entropy maximization framework and, as a result, (1)

is not an integration of the Clausius-Clapeyron equation.

Corrected postprint, 2012-04-16

2

Furthermore, due to the inconsistent assumption, (1) is not accurate enough as an approximation to

be used in practice. Therefore, more accurate empirical relationships are more often used to determine

the saturation vapour pressure. The so-called Magnus-type equations are the most typical in

application and the most recent version [7] for temperature range –40

o

to 50

o

C is

p = 6.1094 exp

+T

T

04.243

625.17 [T in

o

C, p in hPa] (2)

A theoretically consistent closed solution exists [8, p. 203; see also below] and is not much more

complex than (1), but it is rarely mentioned (e.g. none of the above referenced books contains it).

Even when it is mentioned, it is still presented along with (1), which may again be characterized as the

best known approximation to calculate the liquid–vapour equilibrium pressure [9].

Here we demonstrate the inconsistency inherent in the assumptions that are made to derive (1) and

we derive a theoretically consistent closed solution, which is in line with the aforementioned existing

consistent solution, removing incorrect assumptions. We also determine its numerical constants for

the phase change of water, thus providing an expression ready for use in practice, and we show that

numerically its error with respect to standard reference data is negligible, smaller than that of (2) and

spectacularly smaller than that of (1). In all this, the underpinning logic is that the principle of

maximum entropy, which is a variational principle, is economic in assumptions needed:

Mathematically, there is no limit to the number of unknowns that can be determined in a

maximization problem (as compared to one formulated in terms of equations, where the number of

unknowns should equal the number of equations). Thus, we should be able to determine any unknown

quantity without assumptions.

2. The inconsistency

It is well known [e.g. 1, p. 143] that in two systems at equilibrium, entropy maximization constrained

with the conditions that the total energy, volume and number of particles are constant, results,

respectively, in temperature, T, pressure, p, and chemical potential, µ, that are equal in the two

systems. In particular, the last equality is

µ

G

= µ

L

(3)

where the subscripts G and L denote the gaseous and liquid phase, respectively. Classical and

statistical thermodynamics texts do not use (3) directly; rather, they derive, as a consequence of (4),

and use the equality of differentials [e.g. 2. p. 71],

dµ

G

= dµ

L

(4)

The entropies per unit mass s

G

and s

L

at the gaseous and liquid phase, respectively, are given as

s

G

= CT

2

+ c

p

ln T – R ln p (5)

s

L

= CT

3

+ c

L

ln T (6)

where c

p

is the specific heat at constant pressure of the vapour and c

L

is the specific heat of the liquid.

The liquid was regarded as incompressible and, for this reason, in (6) the entropy does not depend on

pressure. The differentials of the chemical potentials are given by Gibbs-Duhem equations [e.g. 2] as

dµ

G

= v

G

dp – s

G

dT (7)

dµ

L

= – s

L

dT (8)

where again in (8) we neglected the specific volume v

L

of the liquid phase. Combining (4), (7) and (8)

we obtain

T

p

d

d

=

G

v

ss

LG

− (9)

If we express the entropy difference in terms of the latent heat as

3

s

G

– s

L

=

T

L

(10)

then we obtain the following typical form of the Clausius-Clapeyron equation,

T

p

d

d

=

G

Tv

L

(11)

Furthermore, using the ideal gas law, pv

G

= RT, we obtain the expression

T

p

d

d

=

2

RT

Lp

(12)

whose integration assuming constant L results in (1).

In fact, however, assuming a constant L renders the above derivation unnecessary. Indeed, we can

express the difference of entropies from (5) and (6) as

s

G

– s

L

= (CT

2

– CT

3

) + (c

p

– c

L

) ln T – R ln p (13)

Substituting into (10) and solving for p we readily obtain

p = CT

4

Rcc

p

T

RT

L

/)(

L

exp

−

− (14)

where CT

4

:= exp[(CT

2

– CT

3

)/R]. Equations (1) and (14) describe the same relationship of p and T,

and were derived by precisely the same assumptions. However they are clearly inconsistent, which

implies that at least one of the assumptions made is excessive and incorrect.

3. Alternative derivation using classical thermodynamics

In this section we will remove the assumption for constant L. We express the difference of entropies

in (13) as an unknown function g(T, p) := s

G

– s

L

and we write (13) in differential form as

(c

p

– c

L

)

T

T

d

– R

p

p

d

= dg (15)

Using the law of ideal gases to eliminate v

G

from (9), we obtain

T

p

d

d

=

RT

p

(s

G

– s

L

) =

RT

pg

(16)

Solving (16) for dp/p and substituting it in (15) we obtain

(c

p

– c

L

)

T

T

d

– g

T

T

d

= dg (17)

which can be written as

(c

p

– c

L

) dT = d(Tg) (18)

and can be readily solved to give

g = s

G

– s

L

=

T

α

– (c

L

– c

p

) (19)

where α is an integration constant. Comparing (19) with (10), we conclude that L appears to be a

linear function of temperature, rather than a constant:

L = α – (c

L

– c

p

)T (20)

Now, substituting g from (19) to (16) we obtain

T

p

d

d

=

RT

p

(c

p

– c

L

) +

2

RT

pα

(21)

4

This is readily solved to give

p = CT

5

Rcc

p

T

RT

α

/)(

L

exp

−

− (22)

Comparing (21) with the earlier results, we observe that it is functionally equivalent with (14) (both

include a multiplicative factor that is a power function of T) whereas (1) proves to have an

inappropriate functional form.

To eliminate CT

5

from (22) we assume a known saturation vapour pressure p

0

at a specific

temperature T

0

. We can then write (22) as

p = p

0

−

T

T

RT

α

0

0

1exp

Rcc

p

T

T

/)(

0

L

−

(23)

which is our final closed solution of the Clausius-Clapeyron equation.

4. Alternative derivation in a statistical mechanical framework

In this section we derive the phase transition equation in a purely statistical mechanical framework

totally avoiding the assumptions about the equality of chemical potentials and temperatures; rather we

will derive them by entropy maximization. For this maximization we assume that our system contains

a total of N particles, N

G

of which in the gaseous phase and N

L

in the liquid phase, so that

N

G

+ N

L

= N (24)

If S denotes the total extensive entropy, and S

G

and S

L

denote the extensive entropy in the gaseous

and liquid phase, respectively, then,

S = S

G

+ S

L

= N

G

s

*

G

+ N

l

s

*

L

(25)

where s

*

G

and s

*

L

denote entropies per particle. From generalized Sackur-Tetrode equations [e.g. 2] we

have

s

*

G

/k = CT

6

+ (c

v

/R) ln (E

G

/N

G

) + ln (V/N

G

), s

*

L

/k = CT

7

+ (c

L

/R) ln (E

L

/N

L

) (26)

where again we neglected the volume per particle in the liquid phase, which is by several orders of

magnitude smaller than that of the gaseous phase. In (26) k is Boltzmann’s constant and c

v

is the

specific heat at constant volume of the vapour. We recall that c

v

= c

p

– R and that the quantity 2c

v

/R

represents the degrees of freedom available to the molecular (thermal) motion. In addition, E

G

and E

L

in (26) are the thermal energies in the two phases. If E is the total energy, then conservation of energy

demands

E

G

+ E

L

+ N

G

ξ = E (27)

where ξ is the amount of energy per molecule required to break the bonds between molecules of the

liquid phase in order for the molecule to move to the gaseous phase, which we assume constant.

We wish to find the conditions which maximize the entropy S in (25) under constraints (24) and

(27) with unknowns E

G

, E

L

, N

G

, N

L

. We form the function Ψ incorporating the total entropy S as well

as the two constraints with Langrage multipliers κ and λ:

Ψ = (S

G

+ S

L

)/k + κ (E

G

+ E

L

+ N

G

ξ – E) + λ (N

G

+ N

L

– N) (28)

To maximize Ψ, we first take the derivatives with respect to E

G

and E

L

and equate them to 0 to obtain

G

Ε

Ψ

∂

∂ =

k

1

G

G

Ε

S

∂

∂ + κ = 0,

L

Ε

Ψ

∂

∂ =

k

1

L

L

Ε

S

∂

∂ + κ = 0 (29)

We recall that in statistical thermodynamics the temperature is defined as

T

1 :=

Ε

S

∂

∂

(30)

5

Thus, (29) results in

κ = –

G

1

kT = –

L

1

kT = –

kT

1 (31)

In other words, it was proved that the temperatures in the two phases are equal.

Furthermore, taking the derivatives of Ψ with respect to N

G

and N

L

, and equating them to 0 we

obtain

G

N

Ψ

∂

∂

=

k

s

*

G

–

R

c

v

– 1 + κξ + λ = 0,

L

N

Ψ

∂

∂

=

k

s

*

L

–

R

c

L

+ λ = 0 (32)

Eliminating λ, substituting κ from (31) and c

v

from c

v

= c

p

– R, and making algebraic manipulations,

we find:

k

ss

*

L

*

G

−

=

kT

ξ

–

R

cc

p

−

L

(33)

On the other hand, from (26), observing that E

G

/N

G

and E

L

/N

L

are both proportional to T, while V/N

G

is proportional to T/p, we also obtain the difference of entropies per particle as:

k

ss

*

L

*

G−

= CT

8

–

R

cc

p

−

L

ln T – ln p (34)

Combining (33) and (34), eliminating s

*

G

– s

*

L

, and solving for p we find

ln p = CT

9

–

kT

ξ

–

R

cc

p

−

L

ln T (35)

Now if we introduce α = ξR/k (= ξN

a

, where N

a

is the Avogadro constant) and take antilogarithms,

then we obtain (22) again, which was our desideratum.

To finish this analysis, we will show the equality of chemical potentials (although the chemical

potential was not involved at all in the above proof). We recall that the chemical potential is by

definition:

–

T

µ

:=

N

S

∂

∂

= s

*

+ N

N

s

∂

∂*

(36)

where the partial derivative applies for constant internal energy. Applying this definition in the two

phases and using (26) we find,

–

T

µ

G

= s

*

G

–

R

kc

v

– k , –

T

µ

L

= s

*

L

–

R

kc

L

+ ξ

Τ (37)

where the last term (ξ/T) in the second equation represents the conversion from constant thermal

energy E

L

to constant internal energy U

L

= E

L

– N

L

ξ. Specifically, this term represents the quantity

(∂S

L

/∂E

L

)(∂E

L

/∂N

L

) = (1/T) ξ. This gives the difference of chemical potentials as

T

µµ

GL −

= s

*

G

– s

*

L

+

R

cck

p

)( L

−

– ξ

Τ (38)

Combining (33) and (38) we find

µ

L

– µ

G

= 0 (39)

5. Application to water vapour

We choose as reference point the triple point of water, for which it is known with accuracy that T

0

=

273.16 K (= 0.01

o

C) and p

0

= 6.11657 hPa [10]. The specific gas constant of water vapour is R =

461.5 J kg

–1

K

–1

. The specific heat of water vapour at constant pressure, again determined at the triple

6

point, is c

p

= 1884.4 J kg

–1

K

–1

and that of liquid water is c

L

= 4219.9 J kg

–1

K

–1

[10], so that c

L

– c

p

=

2335.5 J kg

–1

K

–1

and (c

L

– c

p

)/R = 5.06.

The latent heat at T

0

is L

0

= 2.501 × 10

6

J kg

–1

so that α = L

0

+ (c

L

– c

p

)T

0

= 3.139 × 10

6

J kg

–1

and

ξ / kT

0

= α / RT

0

= 24.9. According to (20), this results in the functional form

L [J kg

–1

] = α – (c

L

– c

p

)T = 3.139 × 10

6

– 2336 T [K] (40)

It can be readily verified that this is very close to a commonly suggested (e.g. [11]) empirical linear

equation for latent heat, i.e.,

L [J kg

–1

] = 3.146 × 10

6

– 2361 T [K] ( = 2.501 × 10

6

– 2361 T

C

[

o

C]) (41)

Figure 1 provides a graphical comparison of equation (40) with (41), as well as with tabulated data

from Smithsonian Meteorological Tables [12], which agree with the equations. Furthermore, it is

important to know that the entropic framework which gives the saturation vapour pressure is the same

framework that predicts the relationship of the latent heat of vaporization with temperature.

2.35

2.4

2.45

2.5

2.55

2.6

2.65

-40 -30 -20 -10 0 10 20 30 40 50

Temperature (°C)

Latent heat (MJ/kg)

Theoretically derived

Observed (tabulated)

Common empirical

Figure 1. Comparison of latent heat of water as given by equation (40) proposed in this

study with the empirical equation (41) and with standard tabulated data from ref. [12].

Now, according to (23), the saturation vapour pressure will be

p = p

0

−

T

T

0

1921.24exp

06.5

0

T

T

, with T

0

= 273.16 K, p

0

= 6.11657 hPa. (42)

where we have slightly modified the last two decimal digits of the constant α / RT

0

to optimize its fit

to the data (see below). For comparison, the inconsistent version (1) for constant L = L

0

is

p = p

0

−

T

T

0

184.19exp

, with T

0

= 273.16 K, p

0

= 6.11657 hPa. (43)

Equations (42) and (43), if plotted on a p vs. T graph, seem indistinguishable from each other as

well as from the Magnus-type equation (2) (Figure 2). However, because p ranges at several orders of

magnitude, the plot of Figure 2 is misleadingly hiding the differences between the different equations.

The maximum relative difference of the proposed equation (42) with respect to (43) exceeds 7%,

while that with respect to (2) is much lower, 0.29%. It is thus more informative to compare the three

equations in terms of relative differences and also to compare them to data rather than to intercompare

to each other.

7

0.1

1

10

100

1000

-40 -30 -20 -10 0 10 20 30 40 50

Temperature (°C)

Saturation vapour pressure (hPa)

Magnus type (by Alduchov & Eskridge)

Clausius-Clapeyron as proposed

Clausius-Clapeyron for constant L

Figure 2. Comparison of saturation vapour pressure obtained by the proposed equation

(42), by the Magnus-type equation (2) from ref. [7], and by the standard but inconsistent

equation (43) for constant L.

-1

0

1

2

3

4

5

6

7

8

-40 -30 -20 -10 0 10 20 30 40 50

Temperature (°C)

Relative difference (%)

Proposed vs. IAPW S Standard vs. IAPW S

Proposed vs. ASHRAE Standard vs. ASHRAE

Proposed vs. Smiths. Standard vs. Smiths.

Proposed vs. W MO Standard vs. WMO

Figure 3. Comparison of relative differences of the saturation vapour pressure obtained

by the proposed equation (42), as well as by the standard but inconsistent equation (43),

with data of different origins (see text).

8

Table 1 Maximum values of the relative differences from data of the saturation vapour

pressure obtained by three different equations (see text).

Standard (inconsistent)

equation (43)

Magnus-type

equation (2)

Proposed

equation (42)

Difference from IAPWS data 6.8% 0.27% 0.07%

Difference from all data 7.6% 0.39% 0.15%

For the comparisons four reference data sets have been used, which are given in tabulated form

from different origins: (a) the International Association for the Properties of Water and Steam

(IAPWS), (b) the Smithsonian Meteorological Tables (Smiths.), (c) the World Meteorological

Organization (WMO) meteorological tables, and (d) the American Society of Heating, Refrigerating

and Air-conditioning Engineers (ASHRAE). The temperature domain of the comparison extends from

–40

o

to 50

o

C, that is the typical range used in hydrometeorological applications. The data set (a),

taken from ref. [10], contains values of saturation vapour pressure for temperatures higher than the

water triple point (273.16 K). The other three data sets, all taken from ref. [13], contain values also

below triple point; such temperatures prevail in the upper air and the saturation vapour pressure in

such temperatures is necessary in order to estimate the relative humidity of the atmosphere. It is

clarified that the values for T < 0

o

C are for water vapour over a surface of liquid water (not over ice),

and thus are relevant to our study. Nonetheless, it is reasonable to expect that the values for T ≥ 0

o

C

are more accurate and that the IAPWS data set, which is newer, is the most accurate among the four.

The different data sets display small differences between each other for the same temperature value,

up to 0.16%.

Figure 3 provides a graphical comparison of equations (42) and (43) with the reference data.

Clearly, the common inconsistent equation (43), derived for constant L, proves to be inappropriate, as

its relative error exceeds 7%. In contrast, the derived closed solution (42) has negligible relative

errors. The maximum relative errors of the two equations, as well as those of the Magnus-type

equation (2), with respect to the data are given in Table 1. It can be seen that the differences of the

proposed equation (42) from the data is negligible, smaller that the deviations among the values of

different data sets. The error of Magnus-type equation (2) is four times larger than that of (42) for the

most accurate IAPWS data set and 2.5 times larger for all data sets. The error of the standard equation

(43) is too high, 1.5-2 orders of magnitude higher that that of the proposed equation (42).

The simplicity of (42) makes numerical calculations easy. For known T, (42) provides p directly.

The inverse problem (to calculate T, i.e. the saturation temperature, also known as dew point, for a

given partial vapour pressure p) cannot be solved algebraically. However, the Newton-Raphson

numerical method at an origin T

0

/T = 1 gives a first approximation T΄ of temperature by

T

T

′

0

= 1 +

06

.

5

921

.

24

1

−

p

p

0

ln (44)

Notably, this is virtually equivalent to solving (43) for T

0

/T. This first approximation can be improved

by re-applying (42) solved for the term T

0

/T contained within the exponentiation, to give

T

T

0

= 1 +

921

.

24

1

p

p

0

ln +

921

.

24

06.5

′

T

T

0

ln

(45)

A systematic numerical investigation showed that a single application of (45) suffices to provide a

value of T with a numerical error in T

0

/T less than 0.1%, while a second iteration (setting the

calculated T as T΄) reduces the error to 0.02%.

6. Summary and concluding remarks

Evidently, theoretically consistent relationships are preferable over purely empirical ones as far as the

former agree with empirical evidence and their use is convenient. The Clausius-Clapeyron equation is

a nice theoretical relationship, but the analytical solution typically contained in books, while it is

9

simple and easy to use, proves to be flawed and also a bad approximation of reality. It is, thus,

reasonable that in calculations of saturation vapour pressure, empirical, typically Magnus-type,

equations are preferred over this theoretical equation. By removing an unnecessary, excessive and

inconsistent assumption that is made within the common derivation, we obtain a closed solution that

is still very simple and combines both theoretical consistency and accuracy. With reference to water

vapour saturation, the proposed solution is by orders of magnitude more accurate than the standard

equation of the literature and also better than the more accurate Magnus-type equation. Compared to

standard tabulated data of saturation vapour pressure for temperature range –40

o

to 50

o

C, which is

relevant to hydrometeorological applications, the derived equation has negligible error. These facts

may allow recommending the use of the derived equation (42) as one combining theoretical

consistency and accuracy.

The alternative theoretical framework proposed, which is based on entropy maximization avoiding

unnecessary assumptions, provides a better understanding and intuition development for the phase

transition. Such understating and intuition development can help to recognize, particularly within the

university education, the power of variational principles and the extremization (maximiza-

tion/minimization) approach over the more common approach that the physical laws are

mathematically expressed only by equations, as well as to recognize the fundamental character of

entropy maximization as a powerful physical principle, contrary to a common perception that physical

laws are only deterministic and mechanistic.

As first indicated by Boltzmann and Gibbs, later succeeded by Shannon who used essentially the

same entropy definition to describe the information content, entropy is none other than a measure of

uncertainty [e.g. 14 , 15 , 16 ]. Thus, the interpretation of the framework proposed is that the

quantification of the phase change relies on maximization of uncertainty. In particular, the entropy in

equations (25) and (26) represents the combined uncertainty as to (a) whether a molecule is in the

liquid or gaseous phase, (b) the molecule’s position in space and (c) the molecule’s kinetic state

expressed by its velocity. This combined uncertainty is maximal at the microscopic (molecular) level.

It is amazing that the entropy maximization represents a principle so powerful as to fully explain and

accurately quantify the phase transition determining its latent heat (equations (20) and (40)) and the

resulting equilibrium vapour pressure (equations (23) and (42)). It is even more amazing that, while at

the microscopic level the uncertainty is maximized, at the macroscopic level all the resulting laws

express near certainties, verified by measurements. This is not a surprise, though, because, given the

probabilistic meaning of entropy, the negligible macroscopic uncertainty can be predicted by

application of probability theory for systems with large number of elements, as the typical

thermodynamic systems are.

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