Clausius equations

Clapeyron-Clausius equation A thermodynamic equation applying to any two-phase equilibrium for a pure substance. The equation states ... 

Clausius-Clapeyron equation See Clapeyron-Clausius equation. 

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... 

This equation is known as the Clapeyron-Clausius equation. (Clapeyron 1834 Clausius 1851.)... 

It is of interest to consider the variation of vapor pressure with temperature. The vapor pressure of a liquid is constant at a given temperature. It increases with increasing temperature upto the critical temperature of the liquid. The liquid is completely in the vapor state above the critical temperature. The variation of the vapor pressure with temperature can be expressed mathematically by the Clapeyron-Clausius equation. Clausius modified the Clapeyron equation in the following manner by assuming that the vapor behaves like an ideal gas. 

This equation is called the Clapeyron-Clausius equation. For many purposes this is rearranged and integrated between the temperature limits Tj and T2, corresponding to the pressure limits P1 and P2,... 

The difference Tm - Tmr is given by the Clapeyron-Clausius equation. Neglecting the variation of the latent heat of melting X (erg/g) and of the derivative dp /dT in the narrow temperature range of interest,... 

Vapour pressure. Vapour pressure (vp) of non associated liquids can be related to temperature (T) using the impirical equation of McGowan (35) which is based on a combination of Trouton s rule and the Clapeyron-Clausius equation as follows -... 

Equation (2) is given in Ref 8, p 161, where it is stated that Berthelot equation is derived from Clausius equation (See item... 

Cj) Clausius Equation of State. It is a modification of vanderWaals equation, in which is introduced an expression l/T(V+c) to correct the pressure-correction term (a) for its variation with temperature. Clausius equation is given in Ref 8, p 161 in the form ... 

The temperature of the adsorbent and vapor with which it is in equilibrium must remain constant during the measurement. With nitrogen as the adsorbate the coolant most often used is liquid nitrogen. The temperature of liquid nitrogen is dependent on ambient pressure and the presence of impurities in the liquid which tend to elevate the boiling point. The Clapeyron-Clausius equation can be used to calculate the change... 

The vap. press, curve of solid iodine is indicated by PO, Fig. 16 that of liquid iodine by 00 and the effect of press, on the m.p. of iodine by ON. At the triple point 0 these curves meet. Fig. 18 shows a similar curve for water. The curve PO thus represents the sublimation curve or hoar-frost line OC. the boiling or vaporization curve, i.e. the effect of press, on the b.p. of the liquid. The same phenomenon occurs with water, iodine, etc., and the principle involved is the same as indicated in the law represented by Clapeyron-Clausius equations with respect to the lowering of the m.p. by an increase of press. Consequently, if the vap. press, of iodine be less than that of th,e triple point, the solid does not melt, but rather sublimes directly without melting at the triple point at 114-15° (89 8 mm.) and A. von Richter at 116 1° (90 mm.). According to R. W. Wood, if the condensation of iodine vapour occurs above —60°, a black granular deposit is formed, but below that temp, a deep red film is produced. 

The heats of fusion and vaporization.—From the lowering of the f.p. of cymene and toluene by the soln. of liquid hydrogen chloride, E. Beckmann and P. Wantig14 calculate the heat of fusion of hydrogen chloride as 10 3 cals, per gram of hydrogen bromide, 7 44 cals. and of hydrogen iodide, 413 cals. D. McIntosh and B. D. Steele calculate from Clapeyron and Clausius equation d log pjdT—XjRT 2, for the mol. ht. 

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... 

Equations ol states of various types of systems are numerous. The Curie equation is the equation of state of a paramagnetic solid. The Beattie and Bridge-man equation. Berthelol equation. Clausius equation. Dieterice equation. Keyes equation, and Van der Waals equation are other examples in this category. 

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 ... 

I] Clausius equation This is vander Waals equation modified by Clausius to account for the variation of vander Waals constant a with-temperature. It is given by,... 

Solution As known, the integrated form of Clapeyron-Clausius equation is,... 

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 ... 

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... 

The total molar polarisation, i.e. that due to 1 gramme-molecule of the compound, is given by the Mosotti-Clausius equation —... 

An equation of fundamental importance which finds extensive application in the one-component, two-phase systems, was derived by Clapeyron and independently by Clausius, from the Second law of thermodynamics and is generally known as the Clapeyron-Clausius equation. The two phases in equilibrium may be any of the following types ... 

Integrated Form of Clapeyron-Clausius Equation for Uquid=Gas Equilibrium... 

The Clapeyron-Clausius equation as applied to liquid = vapour equilibrium, can be easily integrated. The molar volume of a substance in the vapour state is considerably greater than that in the liquid state. In the case of water, for example, the value of Vg at 100 C is 18 1670 = 30060 ml while that of V, is only a little more than 18 ml. Thus, V-Vt can be taken as Vwithout introducing any serious error. The Clapeyron-Clausius equation 1.51, therefore, may be written as... 

The Clapeyron-Clausius equation for solid/ vapour equilibrium may be put as... 

Application of the Clapeyron-Clausius Equation for Solid = Liquid Equilibria... 

The Clapeyron-Clausius equation (1.52) for solid = liquid equilibrium cannot be integrated easily since Vs cannot be ignored in comparison with Vt. Also the laws of liquid state are not so simple as those for gaseous state. However, this equation can be used for calculating the effect of pressure on the melting point of a solid. Eq. 1.52 can also be used for calculating heats of fusion from vapour pressure data obtained at different temperatures. 

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