The Clausius–Clapeyron relation, named after Rudolf Clausius and Benoît Paul Émile Clapeyron, who defined it sometime after 1834, is a way of characterizing a discontinuous phase transition between two phases of matter. On a pressure–temperature (P–T) diagram, the line separating the two phases is known as the coexistence curve. The Clausius–Clapeyron relation gives the slope of this curve. Mathematically,
- 1 Disambiguation
- 2 Derivations
- 3 Approximations
- 4 Applications
- 5 Example
- 6 See also
- 7 References
- 8 Bibliography
The generalized equation given in the opening of this article is sometimes called the Clapeyron equation, while a less general form is sometimes called the Clausius–Clapeyron equation. The less general form neglects the magnitude of the specific volume of the liquid (or solid) state relative to that of the gas state and also approximates the specific volume of the gas state via the ideal gas law.:509
Derivation from state postulate
During a phase change, the temperature is constant, so:508
Using the appropriate Maxwell relation gives:508
Since temperature and pressure are constant during a phase change, the derivative of pressure with respect to temperature is not a function of the specific volume.:57, 62 & 671 Thus the partial derivative may be changed into a total derivative and be factored out when taking an integral from one phase to another,:508
- Δ is used as an operator to represent the change in the variable that follows it—final (2) minus initial (1)
For a closed system undergoing an internally reversible process, the first law is
Using the definition of specific enthalpy, h, and the fact that the temperature and pressure are constant, we have:508
where the shift to capital letters indicates a shift to extensive variables. This last equation is called the Clausius–Clapeyron equation, though some thermodynamics texts just call it the Clapeyron equation, possibly to distinguish it from the approximation below.
When the transition is to a gas phase, the final specific volume can be many times the size of the initial specific volume. A natural approximation would be to replace Δv with v2. Furthermore, at low pressures, the gas phase may be approximated by the ideal gas law, so that v2 = vgas = RT / P, where R is the mass specific gas constant (forcing h and v to be mass specific). Thus,:509
This leads to a version of the Clausius–Clapeyron equation that is simpler to integrate::509
- C is a constant of integration.
These last equations are useful because they relate equilibrium or saturation vapor pressure and temperature to the enthalpy of phase change, without requiring specific volume data. Note that in this last equation, the subscripts 1 and 2 correspond to different locations on the pressure versus temperature phase lines. In earlier equations, they corresponded to different specific volumes and entropies at the same saturation pressure and temperature.
Derivation from Gibbs–Duhem relation
Suppose two phases, I and II, are in contact and at equilibrium with each other. Then the chemical potentials are related by μI = μII. Along the coexistence curve, we also have dμI = dμII. We now use the Gibbs–Duhem relation dμ = − sdT + vdP, where s and v are, respectively, the entropy and volume per particle, to obtain
Hence, rearranging, we have
From the relation between heat and change of entropy in a reversible process δQ = T dS, we have that the quantity of heat added in the transformation is
Combining the last two equations we obtain the standard relation.
Derivation assuming equilibrium
Suppose we have a system in equilibrium, then:
Then assume that p and T are changed, but in such a way that the system is still kept in equilibrium:
By substituting : we get:
which is the Clapeyron equation.
The Clausius-Clapeyron equation is now obtained by inserting the molar volume of an ideal gas into the equation:
An analytical solution to the Clausius-Clapeyron equation, accounting for the weak dependence of the enthalpy of vaporization (or "latent heat") on the temperature, is complex and difficult to use. Fortunately approximations have been developed.
A very good approximation can usually be made using the August-Roche-Magnus formula (usually called the Magnus or Magnus-Tetens approximation, though this is historically inaccurate):
es(T) is the equilibrium or saturation vapor pressure in hPa, which is a function of temperature; T is in Celsius. Since there is only a weak dependence on temperature of the denominator of the exponent, this equation shows that saturation water vapor pressure changes approximately exponentially with T.
Chemistry and chemical engineering
The Clausius–Clapeyron equation for the liquid–vapor boundary may be used in either of two equivalent forms.
- T1 and P1 are a corresponding temperature (in kelvins or other absolute temperature units) and vapor pressure
- T2 and P2 are the corresponding temperature and pressure at another point
- ΔHvap is the molar enthalpy of vaporization
- R is the gas constant (8.314 J mol−1K−1)
This can be used to predict the temperature at a certain pressure, given the temperature at another pressure, or vice versa. Alternatively, if the corresponding temperature and pressure is known at two points, the enthalpy of vaporization can be determined.
The equivalent formulation, in which the values associated with one P,T point are combined into a constant (the constant of integration as above), is
For instance, if the p,T values are known for a series of data points along the phase boundary, then the enthalpy of vaporization may be determined from a plot of ln P against 1 / T.
- As in the derivation above, the enthalpy of vaporization is assumed to be constant over the pressure/temperature range considered
- Equivalent expressions for the solid–vapor boundary are found by replacing the molar enthalpy of vaporization by the molar enthalpy of sublimation, ΔHsub
Meteorology and climatology
Clausius–Clapeyron equations is given for typical atmospheric conditions as
- es is saturation water vapor pressure,
- T is a temperature,
- Lv is latent heat of evaporation,
- Rv is water vapor gas constant.
One of the uses of this equation is to determine if a phase transition will occur in a given situation. Consider the question of how much pressure is needed to melt ice at a temperature ΔT below 0°C. Note that water is unusual in that its change in volume upon melting is negative. We can assume
and substituting in
- L = 3.34×105 J/kg (latent heat of water),
- T = 273 K (absolute temperature), and
- ΔV = −9.05×10−5 m³/kg (change in volume from solid to liquid),
- = −13.5 MPa/K.
To provide a rough example of how much pressure this is, to melt ice at −7 °C (the temperature many ice skating rinks are set at) would require balancing a small car (mass = 1000 kg) on a thimble (area = 1 cm²).
- Van't Hoff equation
- Antoine equation
- ^ a b c d e f g h i Wark, Kenneth (1988) . "Generalized Thermodynamic Relationships". Thermodynamics (5th ed.). New York, NY: McGraw-Hill, Inc.. ISBN 0-07-068286-0.
- ^ a b Çengel, Yunus A.; Boles, Michael A. (1998) . Thermodynamics – An Engineering Approach. McGraw-Hill Series in Mechanical Engineering (3rd ed.). Boston, MA.: McGraw-Hill. ISBN 0-07-011927-9.
- ^ Salzman, William R. (2001-08-21). "Clapeyron and Clausius–Clapeyron Equations". Chemical Thermodynamics. University of Arizona. Archived from the original on 2007-07-07. http://web.archive.org/web/20070607143600/http://www.chem.arizona.edu/~salzmanr/480a/480ants/clapeyro/clapeyro.html. Retrieved 2007-10-11.
- ^ M. G. Lawrence, "The relationship between relative humidity and the dew point temperature in moist air: A simple conversion and applications", Bull. Am. Meteorol. Soc., 86, 225-233, 2005
- ^ Zorina, Yana (2000). "Mass of a Car". The Physics Factbook. http://hypertextbook.com/facts/2000/YanaZorina.shtml.
- M.K. Yau and R.R. Rogers, Short Course in Cloud Physics, Third Edition, published by Butterworth–Heinemann, January 1, 1989, 304 pages. EAN 9780750632157 ISBN 0-7506-3215-1
- J.V. Iribarne and W.L. Godson, Atmospheric Thermodynamics, published by D. Reidel Publishing Company, Dordrecht, Holland, 1973, 222 pages
- H.B. Callen, Thermodynamics and an Introduction to Thermostatistics, published by Wiley, 1985. ISBN 0-471-86256-8
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