Models Methods Software

Dan Hughes

The Clausius-Clapeyron Equation

Much is made about the exponential dependency of the vapor pressure of water as a function of temperature. The implication being, as I understand it, that the Clausius-Clapeyron equation implies that there will be significant additional amounts of water vapor in the atmosphere as the so-called global-average surface temperature increases.

Updated November 27, 2010:
For some reasons unknown to me, this post has gotten a lot of hits. But, as noted back in September, nobody is saying anything. Maybe nobody actually reads the post ?? Maybe it’s completely wrong ??

Updated September 17, 2010. At end of the post.

Updated November 2, 2010:

Here’s a recent review of condensation of a vapor from a non-condensable plus vapor mixture: CONDENSATION FROM A VAPOR-GAS MIXTURE

The Clausius-Clapeyron equation gives the slope of the co-existence curve on the pressure-temperature projection of the equation of state for materials. For these notes, we’ll be interested in the case of a homogeneous mixture of the liquid and vapor phases of a pure substance, water. On the Pressure-Volume plane we engineers call this region of the phase diagram ‘the dome’. The equation strictly holds only for the case of equilibrium mixtures in which the temperature of the liquid and vapor are the same. The equation is also strictly valid only when the interface between the phases is a flat surface without curvature. Basically, the equation is a statement that the Gibbs free energy for the liquid and vapor phases are equal. Along the co-existence curve, the state of the phases is determined by either the pressure or the temperature; given either one of these as an independent variable, the other can be determined, as can all other intensive equation of state properties and thermo-physical ( derivatives of the state properties ) and transport properties.

The equation is useful when modeling and calculating situations that involve changes in the state of a material; for these notes evaporation and condensation of water. However, given the limitations in the previous paragraph, the situations to which it can be applied on a fundamentally rigorous basis are limited. Consider the case of evaporation of water at the atmosphere-liquid interface. Clearly the temperature of the phases are not equal; the state of the liquid is far from its saturation state. And almost all real-world applications will involve interfaces that are not planar. How does the equation get into analyses of, say, evaporation at the interface? For almost all mathematical models of this situation, it will be found that the equation is invoked based on an assumption. A close counting of the number of equations and the eigenvectors pointing into and away from the interface will show that there are not enough equations for the number of unknowns. A model or an assumption is needed in order that the model equation system can be closed. The most straightforward, and easiest, assumption is to take the state of the phases somewhere between the continuous liquid and vapor phases to be the equilibrium saturation state. At this point, the Clausius-Clapeyron equation can be introduced to close the system. Other approximations / assumptions / modeling are of course possible and a wide range is found in the literature and some of these will not need the Clausius-Claperyon equation.

Real World Applications
Consider also that the real-world application is a situation for which the vapor is in fact a part of a mixture of gases, the most simple mixture being air plus the vapor. The presence of the non-condensable gas, air, additionally affects application of the equation to, say, condensation. For condensation under quiescence conditions, no bulk motions for either the liquid or gaseous mixture, the effects of the non-condensable can be fairly understood. The motion of the vapor toward the interface due to the condensation causes the mass fraction of the non-condensable to be higher at the interface than in the bulk of the gaseous mixture. The partial pressure for the vapor is thus higher, and the saturation temperature higher, so that a reduction in the condensation rate is obtained compared to that which would be calculated using the temperature in the bulk vapor-plus-gas mixture. The same approach can be used to determine the effects of the non-condensable gas for the case of evaporation for quiescence conditions.

For real-world applications, quiescence conditions are very likely never attained. The bulk motions of the atmosphere and liquid near the interface, and the consequent topology of the near-interface flows, have important effects on the evaporation and condensation processes. Liquid droplets from sprays due to wave-breaking, for example, present a greatly increased surface area for mass exchange. Uptake of the non-condensable gases, e. g. CO2, by ocean water is very frequently controlled by the microscopic scales of the turbulent motions in the liquid at the interface and not directly related to the bulk motions in either the atmosphere or ocean.

Note that almost none of the liquid of primary interest in modeling and analyses of the climate’s systems, sea water, is a pure substance. And the atmosphere, especially near the interface for the oceans, is not a simple mixture of air-plus-vapor. Physical phenomena and processes other than the Clausius-Clapeyron equation are equally important as, and very likely more important than, the exact representation of the co-existence line. Fidelity of the modeling to the real world must focus on the multitude of physical phenomena and processes that control phase-change in the Earth’s climate systems.

Projections / estimates / of future states of the climate require that all phase-change processes be correctly modeled. That increased atmospheric temperature has a potential to contain increased amounts of water vapor is not the issue. To merely state that the Clausius-Clapeyron equation is sufficient basis of this statement does not address the modeling issues at all. Doesn’t even begin to address those.

Water vapor is heterogeneously distributed within the climate systems. Invoking the Clausius-Clapeyron equation when the deserts of the Earth are of interest does not provide any useful information. It’s very hot there, but there’s almost no water vapor in the atmosphere. If it rains there the water was transported from somewhere else. Some regions are always more or less at the saturation state for water vapor in the atmosphere. The equation doesn’t provide any useful information in this case either. The regions that are prone to highly variable states of water-vapor content in the atmosphere, seasonal monsoon rains, for example, are the critical aspects. In some of these cases, transport of the water content from regions outside the local region is the dominate process. GCMs have yet to be demonstrated to be applicable to useful estimates of the state of the climate at regional level.

Exponential Dependency
Finally, to return to the exponential dependency. At the Earth’s surface we are interested in a limited range of temperature. Very likely, a linear approximation to the Clausius-Clapeyron equation, referenced to a some-what central temperature, will provide sufficient accuracy given the physical phenomena and processes that dominate real-world phase change and the lack of correspondence between the theoretical foundations of the co-existence curve and real-world applications and between the theoretical foundations and the assumptions introduced into the modeling. Given estimates of changes in the temperature, the amount by which atmospheric water-vapor content has the potential to change is a very simple and straightforward calculation.

It is very likely that all these real-world effects are much more important than the relationship given by the Clausius-Clapeyron equation. To simply say the words, ‘Clausius-Clapeyron equation’ so as to imply an understanding of the effects expected to occur under changes in the global-average surface temperature demonstrates a lack of appreciation of the inherent complexity of phase change and mass exchanges in the Earth’s climate systems. The Clausius-Clapeyron equation does not provide any information relative to useful projections / estimates future water-vapor content in the atmosphere. On a so-called global-average basis for the temperature, the potential change in water-vapor content is a small number.


For some unknown reasons this post is getting quite a few hits. But no one is leaving comments.

After writing the post I have found an example that illustrates the problem in a most excellent manner.

This book:

Kevin E. Trenberth (Editor), Climate System Modeling, Cambridge University Press, Cambridge, 1992, Digitally printed version 2009.

has discussions of many aspects of the Earth’s climate systems. Chapter 14 is about Land Surface Process Modeling. Section 14.2, specifically Section 14.2.1, discusses energy fluxes at the atmosphere-land interface. The original modeling of the latent heat flux is stated to be Equation (14.6a). The important aspect is that the equation attempts to use a multiplicative factor, beta, applied to the bulk-to-bulk driving potential and associated resistance.

Following the equation is a discussion of evaporative flux of water vapor from vegetated surface, e. g. leaves. It turns out that leaves have a built-in mechanism that significantly affects the evaporative transfer of water vapor from the interior of the leaves to the atmosphere. This additional resistance cannot be correctly described by the bulk-to-bulk potential approach. Instead, the additional resistance must be explicitly accounted for in the model for evaporative vapor flux.

The situation is directly analogous to the that of condensation ( or evaporation ) as discussed in my original post.


August 10, 2010 - Posted by | Uncategorized |

1 Comment »

  1. Factor in pressure differences from phase changes and the tempero-spatial paths become ‘Why, God, turbulence?’

    Comment by kim | September 24, 2011 | Reply

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