Clausius-Clapeyron equation changes

As pointed out earlier, the equilibrium constant of a system changes with temperature. The form of the equation relating K to T is a familiar one, similar to the Clausius-Clapeyron equation (Chapter 9) and the Arrhenius equation (Chapter 11). This one is called the van t Hoff equation, honoring Jacobus van t Hoff (1852-1911), who was the first to use the equilibrium constant, K. Coincidentally, van t Hoff was a good friend of Arrhenius. The equation is... 

Using the Clausius-Clapeyron Equation Living Graph on the Web site for this book, plot on the same set of axes the lines for AH = 15, 20., 25, and 30. kj-mol 1. Is the vapor pressure of a liquid more sensitive to changes in temperature if AH is small or large ... 

This simple theory is unsatisfactory, in that the rate of change of the difference in free energy of liquid and crystalline lead predicted by the Clausius-Clapeyron equation leads to a temperature scale for Fig. 8 four... 

The slope of the line allows for the determination of the enthalpy of vaporization of water, A//Vap, and the y intercept yields the entropy of vaporization, A. S vap As both the enthalpy and the entropy of water increase as the phase change liquid — vapor occurs, the slope and y intercept of the Clausius-Clapeyron equation are negative and positive, respectively. At 373 K these thermodynamic quantities have values of AHvap = 40.657 kJ mol-1 and ASvap = 109.0 J K-1 mol-1. The leavening action due to water vapor or steam arises from the increased amount of water vapor that forms as pastry temperatures initially rise in the oven and then from the increased volume of the water vapor as temperatures continue... 

All partitioning properties change with temperature. The partition coefficients, vapor pressure, KAW and KqA, are more sensitive to temperature variation because of the large enthalpy change associated with transfer to the vapor phase. The simplest general expression theoretically based temperature dependence correlation is derived from the integrated Clausius-Clapeyron equation, or van t Hoff form expressing the effect of temperature on an equilibrium constant Kp,... 

Having qualitatively discussed the way a pressure cooker facilitates rapid cooking, we now turn to a quantitative discussion. The Clapeyron equation, Equation (5.1), would lead us to suppose that dp oc dT, but the liquid-gas phase boundary in Figure 5.12 is clearly curved, implying deviations from the equation. Therefore, we require a new version of the Clapeyron equation, adapted to cope with the large volume change of a gas. To this end, we introduce the Clausius-Clapeyron equation ... 

The Clausius-Clapeyron equation quantifies the way a boiling temperature changes as a function of the applied pressure. At the boiling points of 7i and T2, the external pressures pi and p2 are the same as the respective vapour pressures. 

We need to understand that the Clausius-Clapeyron equation is really just a special case of the Clapeyron equation, and relates to phase changes in which one of the phases is a gas. 

The Clausius-Clapeyron equation can be used to predict the change in aw with a change in temperature (Kapsalis, 1987) ... 

Since surface pressure is a free energy term, the energies and entropies of first-order phase transitions in the monolayer state may be calculated from the temperature dependence of the ir-A curve using the two-dimensional analog of the Clausius-Clapeyron equation (59), where AH is the molar enthalpy change at temperature T and AA is the net change in molar area ... 

Any one of Equations (8.14), (8.15), or (8.16) is known as the Clausius-Clapeyron equation and can be used either to obtain AH from known values of the vapor pressure as a function of temperature or to predict vapor pressures of a hquid (or a solid) when the heat of vaporization (or sublimation) and one vapor pressure are known. The same equations also represent the variation in the boiling point of a liquid with changing pressure. 

Equation 6.56 is known as the equation of lowering of freezing point and is valid for solid mixtures crystallizing from multicomponent melts. Like the Clausius-Clapeyron equation, it tells us how the system behaves, with changing T, to maintain equilibrium on the univariant curve. However, whereas in the Clausius-Clapeyron equation equilibrium is maintained with concomitant changes in 7) here it is maintained by appropriately varying the activity of the component of interest in the melt and in the solid mixture. 

At fixed (ambient) pressure, a Clausius-Clapeyron equation relates the change in transformation temperature with applied uniaxial load ... 

For pressures below one atmosphere, a number of simplifications can be made. The volume change on formation of vapor can be approximated reasonably by the volume of the vapor. The vapor is assumed to behave like an ideal gas. In particular, AV = VM = RT/ P. Substituting this value for AV into Equation (1) yields the Clausius-Clapeyron equation ... 

The problem with use of the Antoine equation is that its use can introduce unreasonable assumptions about the change in AHv with temperature. This equation tends to overestimate the increase in enthalpy of vaporization with decreasing temperature. Grain (1982) used an approximation to the somewhat more realistic Watson24 expression for this temperature dependence. To calculate the vapor pressure at temperature T, lower than the boiling point, Tb, using the Clausius-Clapeyron equation, Watson suggested the function... 

The integration of Equation (11.22) to determine the equilibrium constant as a function of the temperature or to determine its value at one temperature with the knowledge of its value at another temperature is very similar to the integration of the Clausius-Clapeyron equation as discussed in Section 10.2. The quantity AHB must be known as a function of the temperature. This in turn may be determined from the change in the heat capacity for the change of state represented by the balanced chemical equation with the condition that all substances involved are in their standard states. 

Clausius-Clapeyron equation. Because the magnitude of changes in X and kx in solid... 

Equation (18.52) is the extension of the Clausius-Clapeyron equation to a phase change involving two substances. It gives the influence of temperature on the total vapour pressure of a phase whose composition remains constant. 

The Clausius-Clapeyron equation is not limited to ordinary phase changes, but is applicable to any system whose state can be described in terms of the variables T, p, When applied to the present problem we obtain a formula due to Ehrenfest.f... 

In this case it is easy to show that (29.84) or its equivalent (29.86) can be put in the form of a Clausius-Clapeyron equation. For in the indifferent state considered, the system can undergo an equilibrium transformation without any effect on the composition of the phases. We can show, just as in 2 of chapter XXVIII that this change takes place at constant T andp. The intensive variables v, are thus constant during the transformation, so that on differentiating... 

When the temperature of a liquid is changed from Tj to Tj, the vapor pressure of the liquid changes from Pj to Pj. These changes are related to the molar heat of vaporization, Aff, for the liquid by the Clausius-Clapeyron equation. 

It is well known that the volume of a vapour, V(g), is much larger than that of the equivalent amount of a liquid. As a result, it is reasonable that the V(g) of vapour volume represents the volume change of AVm. Further, the vapour can be considered a perfect gas. Based on the state equation of perfect gases for a mole gas, the Clausius-Clapeyron equation can be rewritten as... 

If the changes in phase equilibria indeed obey Le Chatelier s principle, the system can be specified quantitatively by the Clausius-Clapeyron equation as follows ... 

Salmeterol xinafoate is known to exist in two polymorphic forms, forms I and II. Form I is stable, and form II is the metastable polymorph at ambient temperature. The enthalpies of solution (AHso ) of forms I and II determined from van t Hoff solubility-temperature plots are 32.1 and 27.6 kJ/mol, respectively, and the transition temperature obtained by linear extrapolation of the van t Hoff plots is 99 °C. The enthalpy of polymorphic conversion (AHii i) calculated from the plots of log solubility ratio of polymorphs versus the reciprocal of absolute temperature is negative (—4.55 kJ/mol) (5). However, the change in molar volume (AFu i) due to the conversion is positive. Therefore, according to the Clausius-Clapeyron equation,... 

For the adsorption of a mole of any gaseous molecule onto an inert surface (one that is not changed by the adsorption itself), it can be shown from thermodynamic principles (see Chapter 1) that this vapor-adsorbate phase change is described by the Clausius-Clapeyron equation ... 

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