Chemical potential Clausius-Clapeyron equation

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,... 

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... 

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... 

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 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 ... 

This equation was developed with a different approach, as we saw in Chapter 4, by Clapeyron before the chemical potential was introduced by Gibbs. It is also known as the Clausius-Clapeyron equation, and is often used for the evaluation of the enthalpy of vaporization from vapor pressure and saturated volume data (Section 8.14). 

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