The formula of Clapeyron or relation of Clausius - Clapeyron is a general formula making it possible to calculate the Latent heat L of a phase shift of the matter according to molar volumes of the body in the two phases with balance and knowing the curve of phase shift giving the Pression according to the Température. Or in an opposite way, change of pressure or temperature during a phase shift if the other variables are known.

Definition

The formula is:

$L\; =\; T\; \backslash \; left\; (v\_2\; -\; v\_1\; \backslash \; right)\; \backslash \; frac\; \{dp\}\; \{dT\}\; ~$

This formula is valid in the case of a Transition from phase first order.

(For the transitions from second-order phase to see the Formulas of Ehrenfest)

• $T\; ~$ Temperature to which the phase shift occurs.
• $v\_1,\; v\_2\; ~$ respective mass volume of the body in the two phases with balance.
• $\backslash \; frac\; \{dp\}\; \{dT\}\; ~$ derived from the pressure compared to the temperature.

Demonstration of the formula

We know that when two phases coexist for a pure substance, the chemical Potentiel of this body is the same one in each of the two phases.

$g\_1\; =\; g\_2\; ~$

While differentiating, one obtains:

$\backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; g\_1\}\; \{\backslash \; partial\; T\}\; \backslash \; right)\; \_p\; dT\; +\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; g\_1\}\; \{\backslash \; partial\; p\}\; \backslash \; right)\; \_T\; dp\; =\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; g\_2\}\; \{\backslash \; partial\; T\}\; \backslash \; right)\; \_p\; dT\; +\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; g\_2\}\; \{\backslash \; partial\; p\}\; \backslash \; right)\; \_T\; dp\; ~$

Who is written:

$-\; s\_1\; dT\; +\; v\_1\; dp\; =\; -\; s\_2\; dT\; +\; v\_2\; dp\; ~$

I.e.:

$s\_2\; -\; s\_1\; =\; \backslash \; left\; (v\_2\; -\; v\_1\; \backslash \; right)\; \backslash \; frac\; \{dp\}\; \{dT\}\; ~$

In addition, let us consider the relation of definition of the free Enthalpie:

$G\; =\; H\; -\; TS\; ~$

The two phases being with balance and the temperature being constant, one obtains while differentiating:

$\backslash \; triangle\; H\; =\; T\; \backslash \; left\; (\backslash \; triangle\; S\; \backslash \; right)\; ~$

For a mole of product, one obtains well:

$L\; =\; \backslash \; triangle\; H\; =\; T\; \backslash \; left\; (\backslash \; triangle\; S\; \backslash \; right)\; =\; T\; \backslash \; left\; (s\_2\; -\; s\_1\; \backslash \; right)\; =\; T\; \backslash \; left\; (v\_2\; -\; v\_1\; \backslash \; right)\; \backslash \; frac\; \{dp\}\; \{dT\}\; ~$

Applications

Practical problem

the relation is used to know if there will be phase shift or not according to the conditions of pressure and temperature. For example, it is often used to explain what allows ice-skating on ice: the increase in pressure of the skater dissolves a thin layer of ice under the shoe.

If T = −2 °C, by using Clausius-Clapeyron to calculate the change of pressure necessary, we must have:

$\{\backslash \; Delta\; p\}\; =\; \backslash \; frac\; \{L\}\; \{T\; \backslash \; Delta\; V\}\; \{\backslash \; Delta\; T\}$

While using:

3.34*10⁵ J/kg, T271K, $\backslash \; Delta\; V$ -9.05 *10^-5m3/kg,

and

$\backslash \; Delta\; T$ = 2K,

we obtain

$\backslash \; Delta\; p$ = 27.2 MPa.

It is equivalent to the weight of a fighter of Sumo (mass = 150 kg) being held on a heel switches (surface = 0.5 cm²)!

That does not seem the only reason!

Meteorology

In meteorology, the relation of Clausius-Clapeyron is used usually in the thermodynamic Diagrammes like the Téphigramme S, Skew-T and émagramme S for the calculation of energies of phase shift of atmospheric water. On such a diagram of Pressure versus Temperature (Pt), the line separating the two phases is known like the curve of coexistence: dP / dT .

What is especially important in meteorology is the saturated vapor pressure of the steam $$ e_s and the relation becomes then:

$\backslash \; frac\; \{D\; e\_s\}\; \{dT\}\; =\; \backslash \; frac\; \{L\; e\_s\}\; \{T^\; 2\; R\; \_v\}$ where $R\_v$ is the constant of the Perfect gas S individual (for water 1850 J/Kg-K)

See too

Related articles

• Equation of thermodynamic Clausius
• Diagram
• Diagram of phase

External bonds

• ''' the diagram of the phases of water ''' by [[Weather-France]]

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