Clausius–Clapeyron relation
The Clausius–Clapeyron relation, in chemical thermodynamics, specifies the temperature dependence of pressure, most importantly vapor pressure, at a discontinuous phase transition between two phases of matter of a single constituent.
It is named after Rudolf Clausius[1] and Benoît Paul Émile Clapeyron.
[2] However, this relation was in fact originally derived by Sadi Carnot in his Reflections on the Motive Power of Fire, which was published in 1824 but largely ignored until it was rediscovered by Clausius, Clapeyron, and Lord Kelvin decades later.
[3] Kelvin said of Carnot's argument that "nothing in the whole range of Natural Philosophy is more remarkable than the establishment of general laws by such a process of reasoning.
"[4] Kelvin and his brother James Thomson confirmed the relation experimentally in 1849–50, and it was historically important as a very early successful application of theoretical thermodynamics.
[5] Its relevance to meteorology and climatology is the increase of the water-holding capacity of the atmosphere by about 7% for every 1 °C (1.8 °F) rise in temperature.
is the molar change in enthalpy (latent heat, the amount of energy absorbed in the transformation),
The equation expresses this in a more convenient form just in terms of the latent heat, for moderate temperatures and pressures.
The Clausius–Clapeyron relation describes a Phase transition in a closed system composed of two contiguous phases, condensed matter and ideal gas, of a single substance, in mutual thermodynamic equilibrium, at constant temperature and pressure.
and the total derivative of pressure with respect to temperature may be factored out when integrating from an initial phase
Given that a phase change is an internally reversible process, and that our system is closed, the first law of thermodynamics holds:
Given constant pressure and temperature (during a phase change) and the definition of molar enthalpy
Given constant pressure and temperature (during a phase change), we obtain[8]: 508
from which the derivation of the Clapeyron equation continues as in the previous section.
These last equations are useful because they relate equilibrium or saturation vapor pressure and temperature to the latent heat of the phase change without requiring specific-volume data.
For instance, for water near its normal boiling point, with a molar enthalpy of vaporization of 40.7 kJ/mol and R = 8.31 J/(mol⋅K),
[14] Clapeyron considered a Carnot process of saturated water vapor with horizontal isobars.
For transitions between a gas and a condensed phase with the approximations described above, the expression may be rewritten as [7]
Atmospheric water vapor drives many important meteorologic phenomena (notably, precipitation), motivating interest in its dynamics.
The Clausius–Clapeyron equation for water vapor under typical atmospheric conditions (near standard temperature and pressure) is
This is also sometimes called the Magnus or Magnus–Tetens approximation, though this attribution is historically inaccurate.
[17] But see also the discussion of the accuracy of different approximating formulae for saturation vapour pressure of water.
Under typical atmospheric conditions, the denominator of the exponent depends weakly on
Therefore, the August–Roche–Magnus equation implies that saturation water vapor pressure changes approximately exponentially with temperature under typical atmospheric conditions, and hence the water-holding capacity of the atmosphere increases by about 7% for every 1 °C rise in temperature.
[18] 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
Note that water is unusual in that its change in volume upon melting is negative.
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[19]) on a thimble (area ~ 1 cm2).
This shows that ice skating cannot be simply explained by pressure-caused melting point depression, and in fact the mechanism is quite complex.
[20] While the Clausius–Clapeyron relation gives the slope of the coexistence curve, it does not provide any information about its curvature or second derivative.
