Simple system Helmholtz energy

To establish the molecular thermodynamic model for uniform systems based on concepts from statistical mechanics, an effective method by combining statistical mechanics and molecular simulation has been recommended (Hu and Liu, 2006). Here, the role of molecular simulation is not limited to be a standard to test the reliability of models. More directly, a few simulation results are used to determine the analytical form and the corresponding coefficients of the models. It retains the rigor of statistical mechanics, while mathematical difficulties are avoided by using simulation results. The method is characterized by two steps (1) based on a statistical-mechanical derivation, an analytical expression is obtained first. The expression may contain unknown functions or coefficients because of mathematical difficulty or sometimes because of the introduced simplifications. (2) The form of the unknown functions or unknown coefficients is then determined by simulation results. For the adsorption of polymers at interfaces, simulation was used to test the validity of the weighting function of the WDA in DFT. For the meso-structure of a diblock copolymer melt confined in curved surfaces, we found from MC simulation that some more complex structures exist. From the information provided by simulation, these complex structures were approximated as a combination of simple structures. Then, the Helmholtz energy of these complex structures can be calculated by summing those of the different simple structures. 

Consider a system of N simple spherical particles in a volume V at temperature T. The Helmholtz energy for such a system is... 

We first consider the case of a simple spherical solute s in a very dilute solution in water w. By very dilute, we mean that all solute-solute interactions may be neglected. Formally, this is equivalent to a system containing just one s solute and N water molecules. Also, for convenience, we assume that the system is at a given temperature T and volume V. The Helmholtz energy of solvation of s in this system is... 

Consider a system of N solvent molecules (we use the subscript w for water or any other solvent molecule) and two simple spherical solute molecules at fixed positions Ri and R2, the system being in a volume V and at a temperature T. For simplicity of notation, we reserve the first two indices i = 1,2 for the solutes and the remaining indices i = 3,4,..., N -I- 2 for the solvent molecules. The Helmholtz energy of such a system is given by... 

Primitive models have been very useful to resolve many of the fundamental questions related to ionic systems. The MSA in particular leads to relatively simple analytical expressions for the Helmholtz energy and pair distribution functions however, compared to experiment, a PM is limited in its ability to model electrolyte solutions at experimentally relevant conditions. Consider, for example, that an aqueous solution of NaCl of concentration 6 mol dm (a high concentration, close to the precipitation boundary for this solution) corresponds to a mole fraction of salt of just 0.1 i.e. such a solution is mostly water. Thus, we see that to estimate the density of such solutions accurately the solvent must be treated explicitly, and the same applies for many other thermodynamic properties, particularly those that are not excess properties. The success of the Triolo et approach can be attributed to the incor-... 

For simple fluids, we used an expression similar to (7.7.3), where the first term was the solvation Helmholtz energy of the hard sphere and the second term was the conditional solvation of the soft interaction, given that the hard interaction has been coupled to the system (section 6.14). Here, we substitute (7.7.4) into (7.7.3) to rewrite AA as... 

Once a system has reached equilibrium, its state is independent of how that state was reached. The Gibbs energy is at a minimum if processes at constant T and P are considered. The Helmholtz energy is at a minimum if processes at constant T and V are considered. If a simple closed system is at constant T and P but not yet at equilibrium, a spontaneous process could possibly increase the value of A, but must decrease the value of G. If a simple closed system is at constant T and V but is not yet at equilibrium, a spontaneous process could possibly increase the value of G, but must decrease the value of A. 

The second law of thermodynamics provides the general criterion for spontaneous processes No process can decrease the entropy of the universe. For a closed simple system at constant pressure and temperature the Gibbs energy G cannot increase, and for a closed simple system at constant temperature and volume the Helmholtz energy A cannot increase. 

The molar Helmholtz free energy of mixing (appropriate at constant volume) for such a synnnetrical system of molecules of equal size, usually called a simple mixture , is written as a fiinction of the mole fraction v of the component B... 

This simple relationship allows us to express all the thermodynamic variables in terms of our colloid concentration. The Helmholtz free energy per unit volume depends upon concentration of the colloidal particles rather than the size of the system so these are useful thermodynamic properties. If we use a bar to symbolise the extensive properties per unit volume we obtain... 

In order to determine a system thermodynamically, one has to specify some independent parameters (e.g. N, T, P or V) besides the composition of the system. The most common choice in MC simulation is to specify N, V and T resulting in the canonical ensemble, where the Helmholtz free energy A is the natural thermodynamical potential. However, MC calculations can be performed in any ensemble, where the suitable choice depends on the application. It is straightforward to apply the Metropolis MC algorithm to a simple electric double layer in the iVFT ensemble. It is however, not so efficient for polymers composed of more than a few tens of monomers. For long polymers other algorithms should be considered and the Pivot algorithm [21] offers an efficient alternative. MC simulations provide thermodynamic and structural information, but time-dependent properties are not accessible. If kinetic or time-dependent properties are of interest one has to use molecular dynamic or brownian dynamic simulations. 

To understand robber elasticity we have to revisit some simple thermodynamics (the horror. the horror ). Let s start with the Helmholtz free energy of our piece of rubber, by which we mean that we are considering the free energy at constant temperature and volume (go to the review at the start of Chapter 10 if you ve also forgotten this stuff). If E is the internal energy (the sum of the potential and kinetic energies of all the particles in the system) and 5 the entropy, then (Equation 13-26) ... 

On the lhs of equation (7.238), we have measurable quantities - the solvation Helmholtz (or Gibbs) energy of a solute in H20 and in D20. On the rhs, we have the quantity ((HB) — (HB) ), which measures the change in the average number of HBs in a system of N water molecules, induced by the solvation process. This may be estimated, provided we have an estimated value for the difference sD — aH. Thus, whereas in (7.234) we obtained an estimate of the structure of pure water, here we got an estimate of the structural changes induced by a simple solute. More details can be found in Ben-Naim (1992). 

It is very attractive to couple the 3D-RISM method with the KS-DFT for the electronic structure to self-consistently obtain both classical and electronic properties of solutions and interfaces. The 3D-RISM approach using the 3D-FFT technique naturally combines with the KS-DFT in the planewave implementation. The planewave basis set is convenient for the simple representation of the kinetic and potential energy operators, and is frequently employed for large systems. The hybrid KS-DFT/3D-RISM method is illustrated below by the example of a metal slab immersed in aqueous solvent [28]. In a self-consistent field (SCF) loop the electronic structure of the metal solute in contact with molecular solvent is obtained from the KS-DFT equations modified for the presence of the solvent. The electron subsystem of the interface is assumed to be at the zeroth temperature, whereas its classical counterpart to have temperature T. The energy parameter of the KS-DFT is replaced by the Helmholtz free energy defined as... 

It is convenient, for purposes of graph theory, to imagine a system with the same temperature, volume, and number of particles as the system of interest, but in which the particles do not interact with each other. Such a system is an ideal gas or ideal gas mixture, and its thermodynamic properties have a rather simple form. (Note, however, that if the original system is at high density, the imaginary system is a dense ideal gas. ) The partition function for this ideal gas will be denoted Oig(r, V, Ni,..., its Helmholtz free energy is Ajg(r, V, Ni,..., N ), and the relationship between these two quantities is the same as between A and Q. We now define the quantity pi, p2, .., p ), where... 

Here A - Ajg is the excess Helmholtz free energy with respect to an ideal gas at the same temperature, volume, and number density of each species. Thus, because of the minus sign, the factor kT, and the factor V in the first equality, si can be regarded as a negative dimensionless excess free energy density for the system. Since both A and Aig are extensive thermodynamic properties of the system, A/V and A JV are functions only of the intensive independent variables. Thus si has been expressed as a function of only the temperature and the number density of each species. (Moreover, we have chosen to use j8 = l/Ztr, rather than T, as the independent temperature variable.) It is this quantity si which has a simple representation in terms of graphs, which will be given below. If si can be calculated (exactly or approximately), this leads to (exact or approximate) results for A and hence for all the thermodynamic properties. 

The NFE theory describes a simple metal as a collection of ions that are weakly coupled through the electron gas. The potential energy is volume-dependent but is independent of the position of the electrons. This is valid for both solids and dense liquids. At densities well above that of the MNM transition, we can use effective pair potentials and find the thermophysical properties of metallic liquids with the thermodynamic variational methods usually employed in theoretical treatments of normal insulating liquids. One approach is a variational method based on hard sphere reference systems (Shimoji, 1977 Ashcroft and Stroud, 1978). The electron system is assumed to be a nearly-free-electron gas in which electrons interact weakly with the ions via a suitable pseudopotential. It is also assumed that the Helmholtz free energy per atom can be expressed in terms of the following contributions ... 

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