Clausius-Clapeyron relation: Difference between revisions

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imported>Paul Wormer
(New page: {{subpages}} {{Image|Clapeyron.png|right|300px|The red line in the P-T diagram is the coexistence curve of two phases: I and II. For instance, ''II'' is the vapor and ''I'' the liquid ph...)
 
imported>Paul Wormer
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\frac{\mathrm{d}P}{\mathrm{d}T} = \frac{Q}{T\,(V^{II} - V^{I})} ,
\frac{\mathrm{d}P}{\mathrm{d}T} = \frac{Q}{T\,(V^{II} - V^{I})} ,
</math>
</math>
where  ''Q'' is the molar heat of transition (heat necessary to bring one mole of the compound from phase ''I'' into phase ''II''), ''T'' is the absolute temperature (the abscissa of the point where the slope is computed), ''V''<sup>''I''</sup> is the molar volume of phase ''I'' at the pressure and temperature of the point where the slope is considered and ''V''<sup>''II''</sup> is the same for phase ''II''. The quantity ''P'' is the absolute pressure.  
where  ''Q'' is the molar heat of transition (heat necessary to bring one mole of the compound from phase ''I'' into phase ''II'').  For instance, when phase ''I'' is a liquid and phase
''II'' is a vapor then ''Q'' &equiv; ''H''<sub>v</sub> is the molar [[heat of vaporization]]. Further ''T'' is the absolute temperature (the abscissa of the point where the slope is computed), ''V''<sup>''I''</sup> is the molar volume of phase ''I'' at the pressure and temperature of the point where the slope is considered and ''V''<sup>''II''</sup> is the same for phase ''II''. The quantity ''P'' is the absolute pressure.  


The equation is named after [[Émile Clapeyron]], who derived it around 1834, and [[Rudolf Clausius]].
The equation is named after [[Émile Clapeyron]], who derived it around 1834, and [[Rudolf Clausius]].
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G^{I}(P,T) = G^{II}(P,T). \,
G^{I}(P,T) = G^{II}(P,T). \,
</math>
</math>
The molar Gibbs free energy of phase &alpha; (''I'' or ''II'') is equal to the [[chemical potential]]
The molar Gibbs free energy of phase &alpha; (&alpha; = ''I'', ''II'') is equal to the [[chemical potential]]
&mu;<sup>&alpha;</sup> of this phase. Hence the equilibrium condition can be
&mu;<sup>&alpha;</sup> of this phase. Hence the equilibrium condition can be
written as,
written as,
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phase ''I'' and ''II'', respectively, while the system stays in equilibrium,
phase ''I'' and ''II'', respectively, while the system stays in equilibrium,
:<math>
:<math>
\mu^{I}(P,T)+\Delta \mu^{I}(P,T) = \mu^{II}(P,T)+\Delta \mu^{II}(P,T)
\mu^{I}+\Delta \mu^{I} = \mu^{II}+\Delta \mu^{II}
\;\Longrightarrow\;
\;\Longrightarrow\;
\Delta \mu^{I}(P,T) = \Delta \mu^{II}(P,T)
\Delta \mu^{I} = \Delta \mu^{II}
</math>
</math>
From classical thermodynamics it is known that
From classical thermodynamics it is known that
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</math>
</math>
where ''Q'' is the amount of heat necessary to convert one mole of
where ''Q'' is the amount of heat necessary to convert one mole of
compound from phase ''I'' into phase ''II''. Clearly, when phase ''I'' is liquid and phase
compound from phase ''I'' into phase ''II''. Elimination of the entropy and taking the limit of infinitesimally small changes in ''T'' and ''P'' gives the ''Clausius-Clapeyron
''II'' is vapor then ''Q'' &equiv; ''H''<sub>v</sub> is the molar [[heat of vaporization]]. Elimination of the entropy and taking the limit of infinitesimally small changes in ''T'' and ''P'' gives the ''Clausius-Clapeyron
equation'',
equation'',
:<math>
:<math>
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The Clausius-Clapeyron equation is exact. When we make the following assumptions we may perform the integration:
The Clausius-Clapeyron equation is exact. When we make the following assumptions we may perform the integration:


* The [[molar volume]] of phase ''I'' is negligible compared to the molar volume of phase ''II'', ''V''<sup>''II''</sup> >> ''V''<sup>''I''</sup>
* The [[molar volume]] of phase ''I'' is negligible compared to the molar volume of phase ''II'': ''V''<sup>''II''</sup> >> ''V''<sup>''I''</sup>. In general, far from the [[critical point]], this inequality holds well for liquid-gas transitions.


* Phase ''II''  satisfies the  [[ideal gas law]]
* Phase ''II''  satisfies the  [[ideal gas law]]
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PV^{II} = R T \,
PV^{II} = R T \,
</math>
</math>
* The transition heat ''Q''  is constant over the temperature integration interval. The integration runs from the lower temperature ''T''<sub>1</sub> to the upper temperature ''T''<sub>2</sub> and from ''P''<sub>1</sub> to  ''P''<sub>2</sub>.
:If phase ''II'' is a gas and the pressure is fairly low this assumption is reasonable.
* The transition heat ''Q''  is constant over the temperature integration interval. The integrations run from the lower temperature ''T''<sub>1</sub> to the upper temperature ''T''<sub>2</sub> and from ''P''<sub>1</sub> to  ''P''<sub>2</sub>.


Under these condition the Clausius-Clapeyron equation becomes
Under these condition the Clausius-Clapeyron equation becomes
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\ln\frac{P_2}{P_1} = -\frac{Q}{R}\;\left( \frac{1}{T_2} - \frac{1}{T_1} \right)
\ln\frac{P_2}{P_1} = -\frac{Q}{R}\;\left( \frac{1}{T_2} - \frac{1}{T_1} \right)
</math>
</math>
where ln is the natural (base ''e'') [[logarithm]].
where ln is the natural (base ''e'') [[logarithm]]. We reiterate that for gas-liquid transitions ''Q'' is the [[heat of vaporization]].

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The red line in the P-T diagram is the coexistence curve of two phases: I and II. For instance, II is the vapor and I the liquid phase of the same compound.

The Clausius–Clapeyron relation, is an equation for a phase transition between two phases of a single compound. In a pressuretemperature (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:

where Q is the molar heat of transition (heat necessary to bring one mole of the compound from phase I into phase II). For instance, when phase I is a liquid and phase II is a vapor then QHv is the molar heat of vaporization. Further T is the absolute temperature (the abscissa of the point where the slope is computed), VI is the molar volume of phase I at the pressure and temperature of the point where the slope is considered and VII is the same for phase II. The quantity P is the absolute pressure.

The equation is named after Émile Clapeyron, who derived it around 1834, and Rudolf Clausius.

Derivation

The condition of thermodynamical equilibrium at constant pressure P and constant temperature T between two phases I and II is the equality of the molar Gibbs free energies G,

The molar Gibbs free energy of phase α (α = I, II) is equal to the chemical potential μα of this phase. Hence the equilibrium condition can be written as,

which holds along the red (coexistence) line in the figure.

If we go reversibly along the lower and upper green line in the figure, the chemical potentials of the phases change by ΔμI and ΔμII, for phase I and II, respectively, while the system stays in equilibrium,

From classical thermodynamics it is known that

Here Sα is the molar entropy (entropy per mole) of phase α and Vα is the molar volume (volume of one mole) of this phase. It follows that

From the second law of thermodynamics it is known that for a reversible phase transition it holds that

where Q is the amount of heat necessary to convert one mole of compound from phase I into phase II. Elimination of the entropy and taking the limit of infinitesimally small changes in T and P gives the Clausius-Clapeyron equation,

Approximate solution

The Clausius-Clapeyron equation is exact. When we make the following assumptions we may perform the integration:

  • The molar volume of phase I is negligible compared to the molar volume of phase II: VII >> VI. In general, far from the critical point, this inequality holds well for liquid-gas transitions.
If phase II is a gas and the pressure is fairly low this assumption is reasonable.
  • The transition heat Q is constant over the temperature integration interval. The integrations run from the lower temperature T1 to the upper temperature T2 and from P1 to P2.

Under these condition the Clausius-Clapeyron equation becomes

Integration gives

where ln is the natural (base e) logarithm. We reiterate that for gas-liquid transitions Q is the heat of vaporization.