Clausius equality

Entropy is the ratio of a body s energy to its temperature according to the Clausius equality (as defined in the next section). For a reversible process, the change in entropy is defined by... 

The Clausius equality says that a microscopic process is at equilibrium if dS = dq/T... 

The second Maxwell relation (Equation (4.38)) may remind us of the form of the Clausius equality (see p. 142). Although the first Maxwell relation (Equation (4.37)) is not intuitively obvious, it will be of enormous help later when we look at the changes in G as a function of pressure. 

The pressure is usually calculated in a computer simulation via the virial theorem ol Clausius. The virial is defined as the expectation value of the sum of the products of the coordinates of the particles and the forces acting on them. This is usually written iV = X] Pxi where x, is a coordinate (e.g. the x ox y coordinate of an atom) and p. is the first derivative of the momentum along that coordinate pi is the force, by Newton s second law). The virial theorem states that the virial is equal to —3Nk T. 

If the latter are fixed, the unit of force is defined when a particular value is given to k. If the gram, centimetre, and second are adopted, and k is put equal to 1, then P = 1, so that the unit of force is that force which, acting on a mass of one gram for one second, imparts to it a velocity of one centimetre per second. This unit was called by Clausius a dyne. 

Equation (6) was obtained in a much less direct manner by Clausius in December, 1854, and is Usually known as the Equality of Clausius. It applies only to reversible cycles. 

If the equality of Clausius is applied to a reversible isothermal cycle (T = constant) we obtain ... 

In a Carnot s cycle, the entropy Qi/Ti is taken from the hot reservoir, and the entropy Q2/T2 is given up to the cold reservoir, and no other entropy change occurs anywhere else. Since these two quantities of entropy are equal and opposite, the entropy. change in the hot reservoir is exactly balanced, or, to use an expression of Clausius, is compensated by an equivalent change in the cold reservoir. Again, in any reversible cycle there is on the whole no production of entropy so that all the changes are compensated. 

Referring to Figure 8, temperature Tc is the chamber temperature and Ts is the surface temperature at the salt solution/vapor interface. The temperature of the chamber is well defined and is an experimental variable, whereas Ts must be higher than Tc due to condensation of vapor on the saturated solution surface. We can determine Ts by applying the Clausius-Clapeyron equation to the problem. Assume that the vapor pressures of the surface and chamber are equal (no pressure gradients), indicating that the temperature must be raised at the surface (to adjust the vapor pressure lowering of the saturated solution) to Pc (at Tc) = Ps (at Tc). However, there is a difference in relative humidity between the surface and the chamber, where RHC is the relative humidity in the chamber and RH0 is the relative humidity of the saturated salt solution, and we obtain... 

The latent heat Of vapcrixati o-f ammonium chloride has been determined experimentally by J. C. G. de Marignae21 at atm press, at 33 0 and 43"8 Cals, per mol. this constant has also been calculated from vap. press, data by A. Horstmann and F. M. G. Johnson using Clausius and Clapeyron s equation Tdp/dTfa— v2). In no case is the evidence that the vapour had assumed the equilibrium conditions satisfactory, and A. Smith and R. H. Lombard also apply Clausius and Clapeyron s equation to the measurements of A. Smith and R. P. Calvert of the sat. vap. press, of ammonium chloride. The value of dp/dT was calculated from their vap. press, equation log p=—ajT- -b log T- -c the volume of the solid v2 is negligibly small, and that of the vapour is equal to the reciprocal of the mol. vapour density 1/D. Substituting these values in Clausius and Clapeyron s equation there results s... 

Moreover, if is always positive and nonzero, W must be positive and nonzero except in the case when the two temperatures are equal. This observation results in the Clausius statement of the second law of thermodynamics Heat of itself will not flow from a heat reservoir at a lower temperature to one at a higher temperature. It is in no way possible for this to occur without the agency of some system operating as a heat engine in which work is done by the surroundings on the system. 

A third statement of the second law is based on the entropy. In reversible systems all forces must be opposed by equal and opposite forces. Consequently, in an isolated system any change of state by reversible processes must take place under equilibrium conditions. Changes of state that occur in an isolated system by irreversible processes must of necessity be spontaneous or natural processes. For all such processes in an isolated system, the entropy increases. Clausius expressed the second law as The entropy of the universe is always increasing to a maximum. Planck has given a more general statement of the second law Every physical and chemical process in nature takes place in such a way as to increase the sum of the entropies of all bodies taking any part in the process. In the limit, i.e., for reversible processes, the sum of the entropies remains unchanged. 

The above equations are variously labelled as Clausius-Clapeyron equations. Subject to the satisfactory nature of the assumptions made, a plot (Figure 26.1(a)) of the variation of the natural logarithm of the vapour pressure, In(P/P°), over a liquid measured at various temperatures against the reciprocal of temperature (1 /T) should be linear and have a gradient equal to — Avap H°/R so provides a means of measuring Avap H° for a variety of liquids (Figure 26.1(b)). Also from vapour pressure data for solids at two or more different temperatures one can measure AsubH°. 

It can be seen that, once an assumption is made for the value of M, the only quantity still unknown in the above equation is a, the evaporation coefficient, which must have a finite value equal to or less than 1. If it is assumed that a is constant but unknown, then the vapor pressure at any given temperature is proportional to the vaporization rate, and the enthalpy of vaporization may be found from the Clausius-Clapeyron type treatment. If a value is assigned to a, then vapor pressure values and the entropy of vaporization can be calculated as well. If the entropy of vaporization found in this way is a reasonable value, then the assumed value of a receives support. The latter procedure has been adopted here, and a value of unity has been taken for a. The reasons for choosing this value are ... 

After freezing, the time to sublimate the solvent is given by the drying expressions in Tables 8.3 and 8.4, where the enthalpy of vaporization for drying is replaced by the enthalpy of sublimation. The enthalpy of sublimation is often equal to the sum of the heats of fusion and vaporization [16]. The enthalpy of sublimatian is also substituted for the enthalpy of vaporization in the Clausius Clapeyron equation (8.9) required for the calculation of the solvent partial pressure. The same rate determining steps of boundaiy layer mass transfer and heat transfer as well as pore diffusion and porous heat conduction are applicable in sublimation. 

Hildebrand proposed to compare the values of Ml T at temperatures at which the vapour concentrations are equal for 0-00507 mol/lit. the value is about 27-5 for normal liquids, 32-4 for 1 3, and 32-0 for H2O. He found that the plots of log p (vapour pressure) against log T gave curves having the same gradients at the same vapour concentrations. The approximate Clapeyron-Clausius equation (11a), 7.VIIIL ... 

The pressure p at To will be equal to the pressure p, at Tq—AT, both over a plane liquid surface (pr>Pm at the same temperature, hence for equality, Pr must be at a lower temperature). The Clapeyron-Clausius equation (12), 7.Vni L gives ... 

This is known as the Clausius-Clapeyron equation. If the molar heat of vaporization and the vapor pressure at some temperature are known for a liquid, the vapor pressure at other temperatures can be calculated, provided the assumptions made in the derivation of this equation are valid. Since the normal boiling point of a liquid is defined as the temperature at which tlie vapor pressure equals one aianosphere, it is apparent that only the molar heat of vaporization and the normal boiling point of a liquid need to be known in order to calculate the vapor pressure at other temperatures. 

The principle of Oamot and Clausius then takes the following form If a stem describes a reversible closed cycle, the transformation value , calculated for the whole cycle, is equal to 0. 

Equation (22-19) is useful particularly for pairs of chemically similar liquids. If Raoult s law holds, relative volatility is equal to pjpi- Therefore, it is possible to plot liquid-vapor composition diagrams for closely similar liquids, such as benzene-toluene, without further ado. Note that the value of iJFis not strictly constant over the whole composition range, even for such a mixture, because pi and P2 do not necessarily vary similarly with temperature (Clausius-Clapeyron equation). 

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