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Conservation of Chemical Species
Published in İsmail Tosun, Fundamental Mass Transfer Concepts in Engineering Applications, 2019
A partial molar property is the rate at which an extensive property of the entire mixture, ϕmix, changes with the number of moles of component k in the mixture when temperature, pressure, and number of moles of all components other than k are kept constant. Mathematically, it is expressed as ϕ¯k=(∂ϕmix∂nk)T,P,nj≠k
Partial Molar Properties and Property Changes by Mixing
Published in Juan H. Vera, Grazyna Wilczek-Vera, Classical Thermodynamics of Fluid Systems, 2016
Juan H. Vera, Grazyna Wilczek-Vera
In general, when pure compounds at a pressure P and temperature T are mixed to form a mixture at P and T, the molar property of the mixture is different from the weighed addition of the molar properties of the compounds. When mixed, the compounds contribute to the molar properties of the mixture by their partial molar property values. Thus, for example, the volume change of mixing is Δvm=∑1cxk(v¯k−vk0)=v−∑1cxkvk0
Homogeneous (Single Phase) Mixtures
Published in R. Ravi, Chemical Engineering Thermodynamics, 2020
Thus the partial molar property ui,PM of a component i in a mixture is a measure of the change in the corresponding extensive variable U due to the change in number of moles of i with the temperature, pressure and the mole numbers of all other species kept fixed. We now seek to express ui,PM in terms of the function û and its derivatives.
Computation of partial molar properties using continuous fractional component Monte Carlo
Published in Molecular Physics, 2018
A. Rahbari, R. Hens, I. K. Nikolaidis, A. Poursaeidesfahani, M. Ramdin, I. G. Economou, O. A. Moultos, D. Dubbeldam, T. J. H. Vlugt
For convenience, in this paper, partial molar properties are considered per molecule instead of per mole. In Equation (1), H is the enthalpy of the system, Ni denotes the number of molecules (or mole) of component i, μA is the chemical potential of component A, P is the imposed pressure, T is the temperature, β = 1/(kBT), and kB is the Boltzmann constant. The partial molar volume of component A equals in which V is the volume of the mixture. In molecular simulation, chemical potentials and partial molar properties cannot be computed directly as a function of atomic positions and/or momenta of the molecules in the system [14,15,22–24], and special molecular simulation techniques are required. To date, different molecular simulation techniques have been used to compute partial molar properties: (1) Numerical differentiation (ND): in a multicomponent mixture, a partial molar property of component A is computed directly by numerically differentiating the total property of the mixture at constant temperature and pressure with respect to the number of molecules of component A, while keeping the number of molecules of all other components constant [1,25,26]. This requires several independent and long simulations. Therefore, it is not well suited for multicomponent mixtures. Moreover, the accuracy of the ND depends strongly on the uncertainty of the computed total property [14,15]; (2) Kirkwood–Buff (KB) integrals: Schnell et al. have used KB integrals to compute the partial molar enthalpies for mixtures of gases or liquids [3,25,27–29]. This method uses transformations between ensembles and it is numerically difficult to compute partial molar enthalpies. However, the computation of partial molar volumes using KB integrals is straightforward [30]; (3) Direct method: in their pioneering work in 1987, Frenkel, Ciccotti, and co-workers used the Widom's test particle insertion (WTPI) method [31] to compute partial molar properties of components in a single MC simulation in the NPT ensemble [14,15]. Due to the inefficiency of the WTPI method for high-density systems, application of this method is rather limited [14,32–36]; (4) Difference method (DM): to avoid sampling issues of the WTPI method, an alternative approach was proposed by Frenkel, Ciccotti, and co-workers which uses identity changes between two molecule types [14,15]. From this, partial molar properties of binary systems could be computed. However, if the two molecules are very different in size or have very different interactions with surrounding molecules, identity changes often lead to unfavourable configurations in phase-space, resulting again in poor statistics.