Clapeyron equation vaporization transition

Clausius-Clapeyron Equation. This equation was originally derived to describe the vaporization process of a pure liquid, but it can be also applied to other two-phase transitions of a pure substance. The Clausius-Clapeyron equation relates the variation of vapor pressure (P ) with absolute temperature (T) to the molar latent heat of vaporization, i.e., the thermal energy required to vajxirize one mole of the pure liquid ... 

The temperature variation of ttv may be analyzed by a relationship analogous to the Clapeyron equation to yield the two-dimensional equivalent to the heat of vaporization. The numerical values obtained for this quantity more nearly resemble the bulk values for hydrocarbons than those for polar molecules. This suggests that most of the change in the surface transition involves the hydrocarbon tail of the molecule rather than the polar head. 

Vaporization Transition Clausius-Clapeyron Equation For the liquid-vapor coexistence line ( vapor-pressure curve ), the Clapeyron equation (7.29) becomes... 

The Clapeyron equation does not apply to a continuous transition, since both the entropy (or enthalpy) change and the volume change are zero. For such a transition, in the region of the critical point, the change in the thermodynamic variable given by the second derivative of G can be represented by an exponential equation. For example, in the region of the (vapor + liquid) critical point, AFyap and T are related byp... 

Considering a liquid-vapor transition and I v ip Fliq. we get the approximate Clapeyron equation... 

Besides solid-fluid equilibria, some pure materials can exist in more than one stable solid structure, giving rise to solid-solid equilibria. Examples include equilibria between the fee and bcc forms of iron, equilibria between rhombic and monoclinic sulfur, and equilibria among the many different phases of ice. Such solid-solid phase transitions are accompanied by a volume change and a latent heat, and these two quantities are related through the Clapeyron equation (8.2.27). When a pure material can undergo solid-solid phase transitions, then the substance usually exhibits multiple triple points. Besides the usual solid-vapor-liquid point, the pure substance might also exist in solid-solid-liquid or solid-solid-solid equilibria. Several such triple points occur in water, caused by equilibria involving various forms of ice [13]. 

The vapor pressure equation for the alpha phase is derived by evaluating free energy functions for the solid and the gas at 25 K intervals from 1000 to 1750 K and the transition temperature. For the liquid phase, values are evaluated at 50 K intervals from 1800 to 3600 K and the melting point. For the beta phase, values were evaluated at the transition and melting point temperatures and fitted to the Clausius-Clapeyron equation (Table 15). 

This is the Clapeyron equation. This compact expression makes it possible to predict the change in the coexistence pressure for a given change in coexistence temperature in terms of the easily measurable thermodynamic quantities AH and AE. Equation 9.3 is a general equation governing phase coexistence and can be applied to fusion, vaporization, and sublimation, as well as to solid-solid phase transitions, such as the conversion of diamond and graphite or the fcc-to-bcc transition in iron. 

For phase transition, such as vaporization or subhmation, in which one phase is a gas, the Clapeyron equation can be expressed in a simple approximate form. For such transitions, the molar volume of the gas phase (Vg) is so much greater than that of the hquid or solid (Vuquid or solid) that we can ignore the molar volume of the dense phase in determining the molar volume change for the transition. For example, for the molar volume change on vaporization, we can write... 

Compared to vaporization, the influence of pressure on the melting point is small. In general, the influence of pressure on a phase transition process is given by the Clapeyron equation ... 

The phase diagram shows the location of the phase boundaries between the solid, liquid, and gas phases. All three phases are in equilibrium at the triple point. If the temperature and pressure exceed the so-called critical values, the phase boundary between liquid and vapor vanishes. For this supercritical state, a change of pressure and temperature no longer leads to a change of the state of aggregation. The influence of pressure on a phase transition is given by the Clapeyron equation. 

For liquid-vapor transitions, the Clapeyron equation can be further simplified. In this transition Vmi Tmg- Therefore we may approximate (V g — y ,i) by mg-In this case the Clapeyron equation (7.1.6) simplifies to... 

The same approximation holds for a solid-vapor transition. From Eqs. (5.3-8) and (5.3-12) we obtain the derivative form of the Clausius-Clapeyron equation. For a liquid-vapor transition... 

A special approximate version of the Clapeyron equation is available for transitions from solid to vapor (sublimation) or from liquid to vapor. For these two types of transitions, the volume change is approximately the volume of the gas. This is because the volume of 1 mol of gas is typically very much more than the volume of 1 mol of the substance in its solid or liquid phase. 

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