The fundamental equations of thermodynamics

Now consider a closed system that can alter its volume V. In this case, the work performed by the system is 5FK = pdY- Combining the first and the second laws of thermodynamics for a closed system (i.e. inserting the inequality in Eq. (1.2) into Eq. (1.1)), we obtain 

For any spontaneous change (process) in the system, the inequality given in Eq. (1.3) will be satisfied. The equality will be satisfied only in a reversible process. 

An isolated system is a system that does not exchange work dW = 0, heat dQ = 0, or matter dN = 0 with its surroundings. Consequently, the total internal energy and volume remain constant  

Note that in an isolated system, every spontaneous event that occurs always increases the total entropy. Therefore, at equilibrium, where the properties of a system no longer change, the entropy of the system will be maximized. 

For a system where entropy and volume are held fixed (i.e. dS = 0 and V = 0), a process will occur spontaneously if 

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One of the fundamental equations of thermodynamics concerns systems at equilibrium and relates the equilibrium constant K to the difference in standard free energy (A6°) between the products and the reactants. 

In treating the fundamental equations of thermodynamics, chemical potentials of species are always used, but in making calculations when T and P are independent variables, chemical potentials are replaced by Gibbs energies of formation AfG . Therefore, we will use equation 3.1-10 in the form... 

The fundamental equation of thermodynamics for G for a dilute aqueous solution containing a weak acid HA, and its basic form A- when these species are at equilibrium at a specified pH is (see 4.3-1)... 

Starting from the fundamental equation of thermodynamics for one component ... 

For a reversible process, where the system is always infinitesmally close to equilibrium, the equality in Eq. (1.3) is satisfied. The resulting equation is known as the fundamental equation of thermodynamics... 

From the fundamental equation of thermodynamics, we can deduce relations between the various properties of the system. To see this, let s consider a function f with independent variables x and y. The differential of f (i.e. the total change in /) can be written as ... 

Similarly, if we define the Gibbs free energy G = U — TS + pV, then the fundamental equation of thermodynamics becomes... 

The free energy- that has temperature, volume, and mole numbers as its natural variables is the Helmholtz free energy. Before we stated that once the Gibb s free energy of a system is known as a function of temperature, pressure, and mole numbers G(T,p, N, N2,..all the thermodynamics of the system are known. This is equivalent to the statement that once the Helmholtz free energy is known as a function of temperature, volume, and mole numbers of the system A(T, V, Ni,N2, -all the thermodynamics of the system are known. The fundamental equation of thermodynamics can be written in terms of the Helmholtz free energy as... 

Given and equation of state, we can determine an explicit expression for the Helmholtz free energy from the fundamental equation of thermodynamics. At constant temperature and mole numbers, we have... 

We now regard tlio rafundamental equation of thermodynamics, which includes both the first law and the second. If W is the total energy, S the entropy, T the absolute tenqxu-ature, and F the volume, then, as we know,... 

This combination of the first and second laws of thermodynamics is the fundamental equation of thermodynamics. Using the definitions of the composite functions,... 

Illustrative Problem. Begin with the energy representation of the fundamental equation of thermodynamics U S, V, all Ni) for a system of r-components and transform this complete thermodynamic information to a new state function in which entropy S and all mole numbers Ni [Pg.792]

Illustrative Problem. Begin with the entropy representation of the fundamental equation of thermodynamics for a multicomponent system, S(U, V, all tV,), and derive the Gibbs-Duhem equation. Does this form differ from equation (29-45) in step 4 above ... 

Equations (67a)-(67d) show the particular role of partial molar Gibbs energies Gi. By their definition, quantities G/ are both partial molar quantities and chemical potentials as defined by the fundamental equation of thermodynamics ... 

Electrolyte solutions commonly are one-phase systems of k components. The internal energy U of such systems expressed as a function of its extensive variables S (entropy), V (volume), and n, (amount of substance of component Yi,i = l,2,. ..,k) yields the fundamental equation of thermodynamics. 

The estimates of constants are refined by methods of sequential planning of precise experiments. If necessary, additional chemical investigations are executed in order to come to conclusions about the most probable mechanism. Fishtik and Datta [57,58] have shown that the fundamental equations of thermodynamics have the property of being decomposed into a linear sum of contributions associated with a unique class of reactions referred to as response reactions. This property can also be used for the discrimination among reaction mechanisms. 

Willard Gibbs, after whom the Gibbs energy is named, called Eqs. 5.5.6-5.5.9 the fundamental equations of thermodynamics, because from any single one of them not only the other thermodynamic potentials but also all thermal, mechanical, and chenucal properties of the system can be deduced. Problem 5.4 illustrates this useful application of the total differential of a thermodynantic potential. 

Including the finite interface as a further phase into the fundamental equation of thermodynamics for the free energy F and the free enthalpy G [12],... 

Equation (4) is the fundamental equation of thermodynamic integration (TI). When the derivative of the potential energy with respect to X is known analytically, equation (4) can be applied via a numerical quadrature for a series of X values. In practice, equation (4) has mostly served as the basis of the slow growth (SG) procedure used with MD simulations. Specifically, X is changed incrementally over the full M time steps of the simulation and the average in equation (4) is approximated by a finite difference between the time steps to yield... 

The relations (b)...(e) are called the fundamental equations of thermodynamics. Derivation of these relations had to be limited to reversible processes during which only volume work occurred. However, since internal energy U, enthalpy H, the Gibbs free energy G, and the Helmholtz free energy A are all state functions, the relations derived apply to any change - reversible as well as irreversible - connecting states of equilibrium in a system. Thus, in conclusion we have... 

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