Chemical equations Clausius-Clapeyron equation

Clausius-Clapeyron equation An equation expressing the temperature dependence of vapor pressure ln(P2/Pi) = AHvapCl/Tj - 1/T2)/R, 230,303-305 Claussen, Walter, 66 Cobalt, 410-411 Cobalt (II) chloride, 66 Coefficient A number preceding a formula in a chemical equation, 61 Coefficient rule Rule which states that when the coefficients of a chemical equation are multiplied by a number n, the equilibrium constant is raised to the nth power, 327... 

Most methods for the determination of phase equilibria by simulation rely on particle insertions to equilibrate or determine the chemical potentials of the components. Methods that rely on insertions experience severe difficulties for dense or highly structured phases. If a point on the coexistence curve is known (e.g., from Gibbs ensemble simulations), the remarkable method of Kofke [32, 33] enables the calculation of a complete phase diagram from a series of constant-pressure, NPT, simulations that do not involve any transfers of particles. For one-component systems, the method is based on integration of the Clausius-Clapeyron equation over temperature,... 

For wider temperature ranges, Hv (T) can be expressed as a polynomial or some other function of T. Integration of the Clausius-Clapeyron equation then leads to expressions given in the Handbook of Vapor Pressure (Yaws 1994) or in the Physical and Thermodynamic Properties of Pure Chemicals (Daubert et al. 1994). 

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. 

Chapter 6 dealt with the application of vacuum technology in three areas of the chemical sciences. The first was concerned with its use in chemical technology, particularly in purification/separation operations such as distillation and evaporation. For distillation, the use of the Clapeyron and Clausius-Clapeyron equations was demonstrated (Examples 6.1 and 6.2) whilst Raoult s and Henry s laws were stated and applied (Examples 6.3, 6.4). The removal of water (drying) is an important but poorly understood operation. Aspects of this were discussed in Examples 6.5-6.7. Condensers, particularly in conjunction with vacuum pumps, are indispensable in applications such as distillation and drying. Simple treatment of condenser theory was stated and applied in Examples 6.7-6.9. 

Thermodynamics and kinetics can surely be counted—along with transport phenomena, chemistry, unit operations, and advanced mathematics—as subjects that form the foundation of Chemical Engineering education and practice. Thermodynamics is of course a very old subject. For example, it was the same Rudolf Clausius, who in 1865 coined two immortal sentences (1) "The energy of the universe is constant" and (2) "The entropy of the universe tends to a maximum," that developed the famous Clausius-Clapeyron equation, one of the most basic physico-chemical relationships. Classical thermodynamics was largely complete in the 19th century, before even the basic structure of the atom was understood. 

If the value of p is determined at one temperature, this equation can be solved for Ago, the value of which is needed (along with rot snd vib) to determine the chemical potential of gaseous I2. Once jU-s(7) and /u- (T) are both known, one can calculate AAsub and A//sub. By contrast, the Clausius-Clapeyron equation, given by... 

Estimate the vapor pressure of acetone (mm Hg) at 50 C (a) from data in Perry s Chemical Engineers Handbook and the Clausius-Clapeyron equation, (b) from the Cox chart (Figure 6.1-4), and (c) from the Antoine equation using parameters from Table R4. 

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). 

Sulphur-Halogen Compounds.— The available data for the physicochemical and, primarily, chemical properties of SF have been collated in a review. The variation in the melting point of SF with pressure has been calculated by means of the Clausius-Clapeyron equation as 0.08—0.1Katm which has been confirmed by experiment. The reactions of SFg with various container materials have been studied. From analytical results and i.r. absorption frequencies, the... 

There are several ways to separate an azeotropic mixture into two components of the desired purities, and these are discussed in. other chemical engineering courses. However, one method will be mentioned here, and it is based on the fact that in general the two components will not have the same heat of vaporization, so that by the Clausius-Clapeyron equation the temperature dependence of their vapor pressures will be different. Since the dominant temperature dependence in vapor-liquid equilibrium is that of the pure component vapor pressures, the azeotropic composition will also change with temperature (and pressure). Therefore, what can be done is to use two distillation columns operating at different pressures. 

In some problems, it is not certain in advance what variables should be used for a linear least-squares fit. In the vapor pressure case, we had the Clausius-Clapeyron equation, Eq. (11.43), which indicated that ln(P) and l/T were the variables that should produce a linear relationship. In the analysis of chemical rate data, it may be necessary to try two or more hypotheses to determine which gives the best fit. In a reaction involving one reactant, the concentration c of the reactant is given by Eq.(l 1.45) if there is no back reaction and if the reaction is a first-order reaction. If there is no back reaction and the reaction is a second-order reaction, the concentration of the reactant is given by... 

Here, ASy is the molar entropy of vaporization. Equation (6.85) seems to be closely related to the ebullioscopy law and the law emerging in vapor pressure osmometry. Note that the way of derivation here runs via Raoulfs law and the Clausius - Clapeyron equation, whereas ebullioscopy is derived usually via the chemical potential. Moreover, recalling van l Hoff s law of osmometry, UV = X2RT, we can relate Eq. (6.85) easily to osmometry, arriving at... 

At thermodynamic equilibrium the chemical potential of each component i in both liquid and solid phases has to be equal. For simple systems and certain simplifications, like pure crystalline solid phase of component b (see Walas 1985), thermodynamic considerations lead to the well-known Clausius-Clapeyron equation... 

Vapour pressure p represents the partial pressure of a compound above the pure solid or liquid phase at thermal equilibrium it corresponds to a steady state with a continuous exchange, but no net transfer, of molecules between the two phases. From thermodynamic considerations, the vapour pressure of a chemical is determined by its enthalpy of vaporization (A/f ) and the temperature (7) as described by the Clausius-Clapeyron equation ... 

No validated QSARs are available to predict directly from chemical structure, but there are several methods for calculating p based on derivations of the Clausius-Clapeyron equation (Table 4.4). 

An important physical-chemical property that characterizes the interaction of solid surfaces with gases is the bond energy of the adsorbed species. The determination of bond energy is usually made indirectly by measuring the heat of adsorption (or heat of desorption) of the gas. The heat of adsorption can be determined readily in equilibrium by measuring several adsorption isotherms. The Clausius-Clapeyron equation... 

His essential contributions to Chemical Kinetics, besides the part previously cited in the first part of this chapter, culminated in the discovery of the relation between the rate constant and the equilibrium constant (Van t Hoff, 1884). He interpreted the Chemical Equilibrium as the balance between opvposite reactions so he related equilibrium constant to the ratio of the rate constants of the direct and reverse reaction. From an application of Clausius-Clapeyron equation Van t Hoff found the dependence of the equilibrium constant K from the absolute temperature T ... 

Single-component systems are useful for illustrating some of the concepts of equilibrium. Using the concept that the chemical potential of two phases of the same component must be the same if they are to be in equilibrium in the same system, we were able to use thermodynamics to determine first the Clapeyron and then the Clausius-Clapeyron equation. Plots of the pressure and temperature conditions for phase equilibria are the most common form of phase diagram. We use the Gibbs phase rule to determine how many conditions we need to know in order to specify the exact state of our system. 

The twin problems of cleanliness and structure can now be overcome by the use of single crystals, where both the chemical and physical states of the surface can be monitored using a range of surface spectroscopic techniques. However, single-crystal studies introduce other limitations. In particular the measurements must be carried out under UHV and it is only possible to measure the heats of adsorption indirectly. The most common methods involve either isotherm data and the use of the Clausius-Clapeyron equation or direct analysis of the temperature programmed desorption (TPD) peaks. 

The extent of surface coverage (or simply surface coverage), reached as a result of adsorption, is usually denoted as 0. It is a ratio between adsorbed particles number (Nadi) and the number of adsorption sites available at a surface (usually denoted as active sites - Nsurf). 0 = Nads/ surf The chemical equilibrium between adsorbed species and gas phase particles is reached when chemical potentials of adsorbate particles in both phases are equal (the rates of adsorption and desorption are equal) and it is characterized by constant value of surface coverage 9. The temperature dependence of the gas pressurep required for equilibrium between the adsorption and desorption can be calculated from the Clausius-Clapeyron equation [6], Neglecting the volume of the condensed surface phase, this relation becomes ... 

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