The Clausius-Clapeyron Equation review

 BRIEF HISTORY 

The Clausius-Clapeyron equation is a differential equation that describes the interdependence of pressure and temperature along a pure substance’s phase equilibrium curve. Clapeyron proposed this equation in 1834, and R. Clausius made improvements to it in 1850. In honor of Rudolf Clausius and Benoît Paul Émile Clapeyron, the Clausius Clapeyron equation is a way of explaining a discontinuous phase transformation between two phases of matter of a single constituent. There is no direct relationship between a liquid’s temperature and vapor pressure. The equation may also be referred to as the Clapeyron equation or the Clapeyron-Clausius equation.

Benoît Paul Émile Clapeyron

Rudolf Clausius

 The Clausius-Clapeyron Equation

 We will utilize the Carnot cycle to derive an important relationship, known as the Clausius-Clapeyron Equation or the first latent heat equation. This equation describes how saturated vapor pressure above a liquid change with temperature and also how the melting point of a solid changes with pressure. Let the working substance in the cylinder of a Carnot ideal heat engine be a liquid in equilibrium with its saturated vapor and let the initial state of the substance be T− and es,

 Let the cylinder be placed on a source of heat at temperature T and let the substance expand isothermally until a unit mass of the liquid evaporates. In this transformation the pressure remains constant at es, and the substance passes from state 1 to 2. If the specific volumes of liquid and vapor at temperature T are αl and αv, respectively, the increase in the volume of the system in passing from 1 to 2 is (αv − αl). Also, the heat absorbed from the source is Lv where Lv is the latent heat of vaporization.

 The cylinder is now placed on a nonconducting stand and a small adiabatic expansion is carried out from 2 to 3 in which the temperature falls from T to T − dT and the pressure from es − des.  

The cylinder is placed on the heat sink at temperature T − dT and an isothermal and isobaric compression is carried out from state 3 to 4 during which vapor is condensed.

We finalize by an adiabatic compression from es − des and T − dT to es and T.

 All the transformations are reversible, so We can define the efficiency as in the Carnot Cycle

η = w/ qh = qh − qc /qh = Th − Tc /Th 

And in this specific case of an infinitesimal cycle, we can define the efficiency as in the Carnot Cycle

 dw /qh = dT/ T 

 The work done in the cycle is equal to the area enclosed on a p − V diagram.

Therefore dw = (αv − αl) des 

Also, qh = lv, therefore,

 Lv/ T = (αv − αl) des /dT 

Which can be re-written as

 des /dT = lv/ T (αv − αl) 

Which is the Clausius-Clapeyron Equation

Application of Clausius-Clapeyron equation

• To calculate the slope of a metamorphic reaction using thermodynamic data. This helps to determine whether it might be a geothermometer or geobarometer. Since a geobarometer is more sensitive to pressure changes, it may be a reaction with a shallow dP/dT slope.
• We can also use the equation to calculate the thermodynamic parameters for reactions or phases. When combined with volume results, the slope of an experimentally defined reaction can be used to calculate the S of the reaction and the entropy of formation (Sf) of a specific process. The amounts of phases are frequently well-known, but the data on entropy can be highly uncertain.
• If we have experimental effects on a reaction at one temperature, we can quantify the slope and extrapolate to other conditions instead of conducting more time-consuming tests (or pressure).
• The Clausius Clapeyron equation can correctly position reactions when conducting Schrein makers analysis of an invariant point.
• For conducting comparative research on the characteristics of clouds that form in planetary atmospheres.
• The use of the Clausius-Clapeyron equation thus broadens our understanding of terrestrial water clouds to include various exotic clouds found on other planets.

For further information: CLAUSIUS CLAPEYRON EQUATION || FIRST LATENT HEAT EQUATION || THERMODYNAMICS || WITH EXAM NOTES || - YouTube

References

1.  Clapeyron, E (1834). "Mémoire sur la puissance motrice de la chaleur". Journal de l ́École Polytechnique. XIV: 153–190.
2.  Clausius, R. (1850). "Ueber die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärmelehre selbst ableiten lassen" [On the motive power of heat and the laws which can be deduced therefrom regarding the theory of heat]. Annalen der Physik (in German). 155 (4): 500–524. Bibcode:1850AnP...155..500C. doi:10.1002/andp.18501550403. hdl:2027/uc1.$b242250