The Clausius Equation

Le Chatelier s principle enables us to predict the direction in which the variables of a monovariant system will change under the influence of an imposed change in one of them. By the second law, however, we can obtain more than this merely qualitative result. We can deduce from the second law the amount of the change when the state of the system is given. With this end in view, let us return to the conception of the latent heat of a change of phase in a one-component system as defined on p. 206, viz. 

When the amount dm of the first phase goes into the second phase at constant temperature, the volume of the whole system changes by dv=(v2 v dm, and its energy increases by the amount 

As the system is monovariant, we may write the complete instead of the partial differential coefficient. 

This equation was first obtained by Clapeyron, a French engineer who continued the work of Carnot. He derived the equation for the evaporation of a liquid. In its general form it was established by Clausius, and is therefore called the Clausius-Clapeyron, or briefly the Clausius equation. By means of this equation we can calculate the change in pressure dp produced by an arbitrary change in temperature dT from the quantities L, 2, Vj, and T, which can all be determined by experiment. 

The Clausius equation can be deduced in various ways. According to Helmholtz, the free energy and the total energy of any system are connected by the equation 

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Vacuum evaporation Evaporation at reduced pressure in accordance with the Clausius equation, followed by condensation on a cold surface 1 solid 1 vapour solid on 2 M, Al M2 Fe... 

The advantage over the Clausius equation is that a third empirical constant is not included. 

BERTHELOT EQUATION. A form of the equation of state, relating the pressure, volume, and temperature of a gas, and the gas constant R. The Berthelot equation is derived from the Clausius equation and is of the form... 

The Clausius equation (14) gives the value of the mean free path, L, of a component at ideal conditions, i.e., path without collisions with residual gases ... 

This value of a is now introduced into the integrated form of the Clausius equation, and another equation is obtained which includes the variation of Q with T and which should apply to the whole vapour-pressure curve of liquid phosphorus on the assumption that it is continuous, i.e. that the liquid formed at lower pressures is really the same as that formed at higher pressures. The equations in question are... 

All three derivations of the Clausius equation (3) are identical in principle, as they all make use of the second law of thermodynamics. In giving them all in detail we merely wished to show in what diverse ways the second law can be made to lead to concrete experimental results. The most diverse methods have been employed by various investigators in deriving such results. The choice of method depends partly on the nature of the problem and partly also on the task of the investigator. Van t Hoif, for example, generally used reversible cycles in his classical researches. Other physical chemists prefer Helmholtz s equation or the thermodynamic potential, while partial differential equations, as used in the first of the above derivations, are generally found in physical papers. 

In the following we shall apply the Clausius equation in some specially important cases and compare the theoretical results... 

As the Clausius equation is strictly accurate (in so far as the laws of thermodynamics are accurate), the deviations from experiment, particularly noticeable in the case of water and benzene, must be due to our having neglected quantities which were not negligible, unless, of course, the experimental results are inaccurate. As is always very large compared with at the boiling point, near which all the above data were determined, the deviations must be due to one or both of two causes. Either the vapour does not obey the gas laws, or the latent heat varies... 

The constant coefficients of the serie s can therefore be calculated when the specific heats of vapour and Uquid are known at all temperatures. The integration of the Clausius equation thus depends on this purely experimental problem. 

Application to the process of fvsion. Variation of the melting point with pressure. As a second example of the application of the Clausius equation, let us consider the change of a pure substance from the solid to the liquid state. By the phase rule we have two phases and one component, and hence one degree of freedom. Each pressure wiU therefore correspond to a definite melting point, at which sohd and liquid are in equihbrium. The melting point is therefore a function of the pressure on the system according to the equation... 

Vapour pressure of solutions. The Clausius equation can be applied directly to the evaporation of a solution if we make the restriction that the concentration of the solution shall not alter appreciably when 1 mol. of the solvent is evaporated, which is the case if we are dealing with a very large volume of the solution. The system solution-vapour is then monovariant, and has a definite vapour pressure p at every temperature. If the dissolved substance (solute) has no appreciable vapour pressure, this pressure p is equal to the partial pressure of the solvent. If not, the vapour pressure is equal to the total pressure, i.e. to the sum of the partial pressures of all the components of the solution. In the meantime we shall restrict ourselves to the first case. 

The variation with temperature of the partial pressures and of a solution of composition x can be calculated from the heat of mixture, with the aid of the Clausius equation. 

To derive this equation it is necessary to employ a different method from that used hitherto in this chapter. The Clausius equation can only be applied to variations of pressure with temperature, and is useless where isothermal processes are concerned. We shall, therefore, make use of another consequence of the second law, namely, that no work can be done when two systems in equilibrium with one another are mixed. Thus no work can be done when 1 mol of the component A and x mols of the component B are transferred isothermally from one portion of a solution of composition x to another portion of the same solution. 

The differential heat of solution is of theoretical importance in the calculation of the variation of solubility with pressure. If a saturated solution containing solid solute is subjected to a pressure greater than the vapour pressure of the solution, the gaseous phase disappears and the system becomes divariant (two phases and two components). The concentration of the saturated solution (i.e. the solubility) is then a function of the pressure as well as of the temperature. When a condensed system of this kind is subjected to a further change in pressure the solid solute and the solution will not remain in equilibrium unless the temperature is changed simultaneously. As in the analogous case of the variation with pressure of the melting point of a pure substance (p. 221), the Clausius equation assumes the form — L/ ... 

These equations are called the Clausius equations although the second, which relates the heat capacity change to the temperature coefficient of the heat of reaction, is also called Kirchhoff s equation and the last equation, giving the effect of volume on the heat of reaction, was first derived by De Donder. 

Now let us see to what conclusions the Clausius Equation leads when treated in the same manner This equation, it will be remembered, differs from van der Wails or Ramsay and Young s m that the cohesive force was considered to vary with the temperature, and at the same time was a more complex function of the density The equation is—... 

Symbolic determination of enthalpy departure function for the Clausius equation of state... 

Does a fluid obeying the Clausius equation of state have a vapor-liquid transition ... 

The first item presents the entropy change of the system at the expense of its flux through the boundary and if the fluxes of substance through the boundary are absent, it is given by the Clausius equation ... 

From this principle, and, from the Clausius equation (13), it follows that... 

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