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Clausius-Clapeyron relation
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The Clausius-Clapeyron relation, named after Rudolf Clausius and Émile Clapeyron, who defined it sometime after 1834, is a way of characterizing the phase transition between two phases of matter, such as solid and liquid. 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,
where is the slope of the coexistence curve, is the latent heat, is the temperature, and is the volume change of the phase transition.
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.
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Encyclopedia
The Clausius-Clapeyron relation, named after Rudolf Clausius and Émile Clapeyron, who defined it sometime after 1834, is a way of characterizing the phase transition between two phases of matter, such as solid and liquid. 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,
where is the slope of the coexistence curve, is the latent heat, is the temperature, and is the volume change of the phase transition.
Disambiguation
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.
Derivation
Using the state postulate, take the specific entropy, , for a homogeneous substance to be a function of specific volume, , and temperature, .
During a phase change, the temperature is constant, so
.
Using the appropriate Maxwell relation gives
.
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. Thus the partial derivative may be changed into a total derivative and be factored out when taking an integral from one phase to another,
,
.
- 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, , and the fact that the temperature and pressure are constant, we have
.
After substitution of this result into the derivative of the pressure, one finds
,
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 with . Furthermore, at low pressures, the gas phase may be approximated by the ideal gas law, so that , where R is the mass specific gas constant (forcing and to be mass specific). Thus,
.
This leads to a version of the Clausius-Clapeyron equation that is simpler to integrate:
,
, or
.
- is a constant of integration
These last equations are useful because they relate saturation pressure and saturation 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.
Other derivation
Suppose two phases, I and II, are in contact and at equilibrium with each other. Then the chemical potentials are related by . Along the coexistence curve, we also have . We now use the Gibbs-Duhem relation
, where and 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 dQ = T dS, we have that the quantity of heat added in the transformation is
Combining the last two equations we obtain the standard relation.
Applications
Chemistry and Chemical Engineering
The Clausius-Clapeyron equation for the liquid-vapor boundary may be used in either of two equivalent forms.
where
- and are a corresponding temperature (in kelvin or other absolute temperature units) and vapor pressure
- and are the corresponding temperature and pressure at another point
- is the molar enthalpy of vaporization
- 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 against .
Notes:
- 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,
Clausius-Clapeyron equations is given for typical atmospheric conditions as
where:
- is saturation water vapor pressure
- is a temperature
- is latent heat of evaporation
- is water vapor gas constant.
Example
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 below 0 °C. Note that water is unique in that its change in volume upon melting is negative. We can assume
and substituting in
- = 3.34 J/kg (latent heat of water),
- = 273 K (absolute temperature), and
- = -9.05 m³/kg (change in volume from solid to liquid),
we obtain
= -13.1 MPa/°C.
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).
See also
Bibliography
- 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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