Clapeyron s equation

Barrer s discussion4 of his analog of Eq. 28 merits some comment. Equation 28 expresses the equilibrium condition between ice and hydrate. As such it is valid for all equilibria in which the two phases coexist and not only for univariant equilibria corresponding with a P—7" line in the phase diagram. (It holds, for instance, in the entire ice-hydratell-gas region of the ternary system water-methane-propane considered in Section III.C.(2).) In addition to Eq. 28 one has Clapeyron s equation... 

From Clapeyron s equation we have and if we combine this with the relations... 

Antoine s and Clapeyron s equations offer two possible solutions, and there is also the model developed by Hass and Newton. ... 

The study is based on four iinear hydrocarbons (in Ci, Ce to Ca) and the model uses Antoine and Clapeyron s equations. The flashpoints used by the author do not take into account all experimental values that are currently available the correlation coefficients obtained during multiple linear regression adjustments between experimental and estimated values are very bad (0.90 to 0.98 see the huge errors obtained from a correlation study concerning flashpoints for which the present writer still has a coefficient of 0.9966). The modei can be used if differences between pure cmpounds are still low regarding boiling and flashpoints. 

The melting of bromine or of iodine is attended by an expansion—with bromine, J. I. Pierre 31 found a 6 per cent, expansion. M. Toepler found an expansion of 0 0511 c.c. per gram of bromine, and 0 0434 c.c. per gram of iodine. Hence, by Clausius and Clapeyron s equation the m.p. of bromine is raised 0 0203° per atm. rise of press., and iodine, 0-0314° per atm. rise of press. 

The tendency of the lithium salts to form complexes with ammonia is likewise shown by the partition of ammonia between chloroform and aq. soln. of the lithium salt.78 Dry salts of lithium also form complexes with ammonia, and a comparison of the observed thermal value of the reaction with that computed from the dissociation press, of Clapeyron s equation has been made by J. Bonnefoi, and indicated in Table XXI. According to F. Ephraim, lithium tetrammino-chloride has a vap. press, of 760 mm. at 12°. 

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... 

We can write Clapeyron s equation in a more compact notation by letting phase transition (i.e., the constraint of remaining on the coexistence curve of a, phases)... 

Equation (3) is applicable to various equilibria such as solid-liquid equilibria, liquid-vapour equilibria and equilibria between two solid modifications. The Clapeyron s equation for these various equilibria can be easily obtained as follows ... 

Starting from this concept, let us estimate the concentration and pressure of H2 molecules in the van-der-Waals gap of HJnSe at x = 2. As the volume of y-InSc conventional hexagonal cell is equal V -351 A3, the concentration of elementary cells is N0 = l/ V = 2.85-1021 cm 3. At x = 2, one H2 molecule corresponds to one In2Se2 molecule. Hence, H2 molecular concentration is equal N = 37V0= 8.55 x 1021 cm"3. Using Clapeyron s equation for the ideal gas pressure P = NkBT, where kB is the Boltzmann constant and T is the absolute temperature, we can deduce that the pressure caused by H2 molecules in InSe van-der-Waals gap is equal to 9.4 MPa at... 

The Latent Heats and Clapeyron s Equation.—There is a very important thermodynamic relation concerning the equilibrium between phases, called Clapeyron s equation, or sometimes the Clapeyron-Clausius equation. By way of illustration, let us consider the vaporization of water at constant temperature and pressure. On our P-V-T surface, the process we consider is that in which the system is carried along an isothermal on the ruled part of the surface, from the state whore it is all liquid, with volume Fz, to the state where it is all gas, with volume F . As we go along this path, we wash to find ihe amount of heat absorbed. We can find this from one of Maxwell s relations, Eq. (4.12), Chap. II ... 

Clapeyron s equation holds, as we can see from its method of derivation, for any equilibrium between phases. In the general ease, the difference of volumes on the right side of the equation is the volume after absorbing the latent heat L, minus the volume before absorbing it. 

There is another derivation of Clapeyron s equation which is very instructive. This is based on the use of the Gibbs free energy G. In the last section we have seen that this quantity must be equal for two phases in equilibrium at the same pressure and temperature, and that if one phase has a lower value of G than another at given pressure and temperature, it is the stable phase and the other one is unstable. Wo can verify these results in an elementary way. We know that in going from liquid to vapor, the latent heat L is the difference in enthalpy between gas and liquid, orL = // — Hi. Bui if the change is carried out in equilibrium, the heat absorbed will also equal T (IS, so that the latent heat will be T(SU — Si). Equating these values of the latent heat, we have ... 

Clapeyron s equation, as an exact result of thermodynamics, is useful in several ways. In the first place, we may have measurements of the equation of state but not of the latent heat. Then we can compute the latent heat. This is particularly useful for instance at high pressures. 

The Integration of Clapeyron s Equation and the Vapor Pressure Curve.—The integration of Clapeyron s equation to got the vapor pressure curve over a liquid or solid from a measurement of the latent heat is one of its principal uses. We may write the integral of Eq. (4.3) in the form... 

We can think of two limiting sorts of transitions one in which the transition always occurs at the same temperature independent of pressure, the other where it is always at the same pressure independent of temperature. These would correspond to vertical and horizontal lines respectively in Fig. XI-3. In Clapeyron s equation dP/dT = L/TAV, these correspond to the case dP/dT = oo or 0 respectively. Thus in the first case we must have AV = 0, or the two phases have the same volume, in which case pressure does not affect the transition. And in the second case L = 0, or AS = 0, there is no latent heat, or the two phases have the same entropy, in which case temperature does not affect the transition. Put differently, increase of pressure tends to favor the phase of small volume, increase of temperature favors the phase of large entropy. 

The relation between the temperature and the equilibrium pressure in the one-component system and two phases is defined by the Clapeyron s equation... 

This equation also follows directly from the generalised form of Clapeyron s equation, (7), 41.11. It was first given by Lord Kelvin (1858). Since in all... 

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