The chemical potentials

The thermodynamic theory which has been developed so far is applicable, as it stands, only to closed systems. This applies, for example, to the important relations 

These were based on the discussion of a body which was supposed not to exchange material with its environment. In general, these relations are incorrect, or at least ambiguous,f if an exchange of matter takes place (but they may always be applied to the total system comprising the body and its environment). 

The same remarks apply also to the fundamental equations of the last section. These are not applicable to open systems, or to closed systems which undergo irreversible changes of composition. Consider, for example, the equation 

These difSculties are due to the assumption, implicit in most of Chapter 1, that two variables alone may be sufficient to fix the state of a system. This can only be true for bodies of fixed composition. We must therefore introduce into the equations the variables which determine the composition and size of the system, i.e. the mole numbers. 

Consider a homogeneous phase in which there are h different substances. Let (mols) be the amount of substance 1 in the whole of the phase, (mols) the amount of substance 2, etc. According to (2 34) if 9h,n2, constant, the internal energy U of the 

The pressure equation is very useful in computing the equation of state of a system based on the knowledge of the form of the function g(R). Indeed, such computations have been performed to test theoretical methods of evaluating g(R). 

In a mixture of c components, the generalization of (3.37) is straightforward. Instead of the density p in the first term on the rhs of (3.37), we use the total density pT = J2, Pi- Also, the second term is replaced by a double sum over all pair of species. The result is 

For a system of rigid, nonspherical molecules, the derivation of the pressure equation is essentially the same as that for spherical molecules. The result is 

The chemical potential is the most important quantity in chemical thermodynamics and, in particular, in solution chemistry. There are several routes for obtaining a relationship between the chemical potential and the pair correlation function. Again we start with the expression for the chemical potential in a one-component system, and then generalized to multicomponent systems simply by inspection and analyzing the significance of the various terms. 

In this section, we discuss several different routes to build up the expression for the chemical potential. Note, however, that in actual applications only differences in chemical potentials can be measured. 

To assess the spontaneity of a biological process involving a mixture, we need to know how to compute the contribution of each substance to the total Gibbs energy of the mixture. 

Similarly, by the use of Equation (4.24) and the knowledge that the temperature of the system is uniform, 

We have seen in the preceding sections that the chemical potentials are extremely important functions for the determination of equilibrium relations. Indeed, all of the relations pertaining to the colligative properties of solutions are readily obtained from the conditions of equilibrium involving the chemical potentials. In many of the relations developed in the remainder of this chapter the chemical potentials appear as independent variables. It would therefore be extremely convenient if their values could be determined by direct experimental means. Unfortunately, this is not the case and we must consider them as functions of other variables. 

From their definitions (Eq. (4.39)) based on the energy, the enthalpy, and the Gibbs and Helmholtz energies, we may set the chemical potentials to be functions of other variables, as follows  

The chemical potential is the key property in the most important application of chemical engineering thermodynamics, chemical and phase equilibrium. The ease of separation, for example, of methanol from its mixture with water by distillation is determined by the relationship between the concentration of methanol in the liquid and vapor phases, which - as Gibbs showed (Section 4.12) - is dictated by the chemical potential of methanol in the two phases. 

We saw in Section 9.4, Eq.9.4.16, that the chemical potential is defined in terms of all four energy functions. Since, however, we are interested in chemical and in phase equilibrium reached at constant temperature and pressure, we use the one through the Gibbs free energy. For a pure compound - of interest in this Chapter - Eq.9.4.15 becomes  

the chemical potential is die Gibbs free energy per mole. It follows then from Eq.9.3.19 that  

To evaluate the chemical potential at some temperature T and pressure P, n(T,P), we integrate Eq.9.9.2  

This uncertainty is made worse by the following observation. Assume, for simplicity, that we  

It is sometimes convenient to refer to thermodynamic values for precisely Imol of a substance. The enthalpy, internal energy, entropy, and so on can be expressed per mole. An overbar is used to identify such molar values, and the general definition is 

the derivative of the enthalpy with respect to the number of moles of a substance, is the molar enthalpy. For systems involving different phases of one or of several substances, a set of variables, the composition variables, n, n. .. give the number of moles of each substance in a phase, and then the definition of a molar value becomes 

Equation 4.8a is simply Equation 4.8b applied to a system with one pure substance however, the values defined by Equation 4.8b are usually referred to as partial molar values. It follows from this definition that a total thermodynamic value X for a system can be expressed as a sum of products of partial molar values and the corresponding n, values. 

For example, the internal energy of ice and water in equilibrium is the sum of the partial molar internal energy of ice times the number of moles of ice plus the partial molar internal energy of liquid water times the number of moles of liquid. 

The partial molar Gibbs energy, G has special utility such that it is usually given a special name, the chemical potential, and a distinct symbol, which herein is gi g = G). Equation 4.9 for G is 

For reasons that will become clear in the following paragraphs, the chemical potential cannot be expressed as a simple integral involving the pair correlation function. 

Consider, for example, the pressure equation that we have derived in the previous section, which we denote by 

By this notation, we simply mean that we have expressed the pressure as a function of q and T, and also in terms of g R) (which is itself a function 

We see that in order to express a in terms of g R), we must know the explicit dependence of g R) on the density. Thus, even if we have used the pressure equation in the integrand of (3.47), we need a second integration, over the density, to get the Helmholtz free energy per particle. The chemical potential follows from the relation 

As a second route to computing a, consider the energy equation derived in Section 3.3, which we denote by 

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The chemical potential pi, has been generalized to the electrochemical potential Hj since we will be dealing with phases whose charge may be varied. The problem that now arises is that one desires to deal with individual ionic species and that these are not independently variable. In the present treatment, the difficulty is handled by regarding the electrons of the metallic phase as the dependent component whose amount varies with the addition or removal of charged components in such a way that electroneutrality is preserved. One then writes, for the ith charged species. 

The electrochemical potentials pi, may now be expressed in terms of the chemical potentials pt, and the electrical potentials (see Section V-9) ... 

The electrochemical potential is defined as the total work of bringing a species i from vacuum into a phase a and is thus experimentally defined. It.may be divided into a chemical work p , the chemical potential, and the electrostatic work ZiC0 ... 

Thus Pi, surface potential jump X, the chemical potential p, and the Galvani potential difference between two phases A0 = are not. While jl, is defined, there is a practical dif-... 

Using the temperature dependence of 7 from Eq. III-l 1 with n - and the chemical potential difference Afi from Eq. K-2, sketch how you expect a curve like that in Fig. IX-1 to vary with temperature (assume ideal-gas behavior). 

It is generally assumed that isosteric thermodynamic heats obtained for a heterogeneous surface retain their simple relationship to calorimetric heats (Eq. XVII-124), although it may be necessary in a thermodynamic proof of this to assume that the chemical potential of the adsorbate does not show discontinu-... 

In an irreversible process the temperature and pressure of the system (and other properties such as the chemical potentials to be defined later) are not necessarily definable at some intemiediate time between the equilibrium initial state and the equilibrium final state they may vary greatly from one point to another. One can usually define T and p for each small volume element. (These volume elements must not be too small e.g. for gases, it is impossible to define T, p, S, etc for volume elements smaller than the cube of the mean free... 

Here p is the chemical potential just as the pressure is a mechanical potential and the temperature Jis a thennal potential. A difference in chemical potential Ap is a driving force that results in the transfer of molecules tlnough a penneable wall, just as a pressure difference Ap results in a change in position of a movable wall and a temperaPire difference AT produces a transfer of energy in the fonn of heat across a diathennic wall. Similarly equilibrium between two systems separated by a penneable wall must require equality of tire chemical potential on the two sides. For a multicomponent system, the obvious extension of equation (A2.1.22) can be written... 

The chemical potential now includes any such effects, and one refers to the gmvochemicalpotential, the electrochemical potential, etc. For example, if the system consists of a gas extending over a substantial difference in height, it is the gravochemical potential (which includes a tenn m.gh) that is the same at all levels, not the pressure. The electrochemical potential will be considered later. 

Equation (A2.1.23) can be mtegrated by the following trick One keeps T, p, and all the chemical potentials p. constant and increases the number of moles n. of each species by an amount n. d where d is the same fractional increment for each. Obviously one is increasing the size of the system by a factor (1 + dQ, increasing all the extensive properties U, S, V, nl) by this factor and leaving the relative compositions (as measured by the mole fractions) and all other intensive properties unchanged. Therefore, d.S =. S d, V=V d, dn. = n. d, etc, and... 

Equation ( A2.1.39) is the generalized Gibbs-Diihem equation previously presented (equation (A2.1.27)). Note that the Gibbs free energy is just the sum over the chemical potentials. 

In experimental work it is usually most convenient to regard temperature and pressure as die independent variables, and for this reason the tenn partial molar quantity (denoted by a bar above the quantity) is always restricted to the derivative with respect to Uj holding T, p, and all the other n.j constant. (Thus iX = [right-hand side of equation (A2.1.44) it is apparent that the chemical potential... 

On the other hand, in the theoretical calculations of statistical mechanics, it is frequently more convenient to use volume as an independent variable, so it is important to preserve the general importance of the chemical potential as something more than a quantity GTwhose usefulness is restricted to conditions of constant temperature and pressure. 

In passing one should note that the metliod of expressing the chemical potential is arbitrary. The amount of matter of species in this article, as in most tliemiodynamics books, is expressed by the number of moles nit can, however, be expressed equally well by the number of molecules N. (convenient in statistical mechanics) or by the mass m- (Gibbs original treatment). 

Note that a constant of integration p has come mto the equation this is the chemical potential of the hypothetical ideal gas at a reference pressure p, usually taken to be one ahnosphere. In principle this involves a process of taking the real gas down to zero pressure and bringing it back to the reference pressure as an ideal gas. Thus, since dp = V n) dp, one may write... 

Given this experimental result, it is plausible to assume (and is easily shown by statistical mechanics) that the chemical potential of a substance with partial pressure p. in an ideal-gas mixture is equal to that in the one-component ideal gas at pressure p = p. 

For precise measurements, diere is a slight correction for the effect of the slightly different pressure on the chemical potentials of the solid or of the components of the solution. More important, corrections must be made for the non-ideality of the pure gas and of the gaseous mixture. With these corrections, equation (A2.1.60) can be verified within experimental error. 

It follows that, because phase equilibrium requires that the chemical potential p. be the same in the solution as in the gas phase, one may write for the chemical potential in the solution ... 

Instead of using the chemical potential p. one can use the absolute activity X. = exp( xJRT). Since at equilibrium A= 0,... 

To proceed fiirther, to evaluate the standard free energy AG , we need infonnation (experimental or theoretical) about the particular reaction. One source of infonnation is the equilibrium constant for a chemical reaction involving gases. Previous sections have shown how the chemical potential for a species in a gaseous mixture or in a dilute solution (and the corresponding activities) can be defined and measured. Thus, if one can detennine (by some kind of analysis)... 

This is the same as that in the canonical ensemble. All the thennodynamic results for a classical ideal gas tlien follow, as in section A2.2.4.4. In particular, since from equation (A2.2.158) the chemical potential is related to which was obtained m equation (A2.2.88). one obtains... 

In an ideal Bose gas, at a certain transition temperature a remarkable effect occurs a macroscopic fraction of the total number of particles condenses into the lowest-energy single-particle state. This effect, which occurs when the Bose particles have non-zero mass, is called Bose-Einstein condensation, and the key to its understanding is the chemical potential. For an ideal gas of photons or phonons, which have zero mass, this effect does not occur. This is because their total number is arbitrary and the chemical potential is effectively zero for tire photon or phonon gas. 

The chemical potential for an ideal Bose gas has to be lower than the ground-state energy. Otherwise the occupancy (n.p of some state j would become negative. 

Fluctuations of observables from their average values, unless the observables are constants of motion, are especially important, since they are related to the response fiinctions of the system. For example, the constant volume specific heat of a fluid is a response function related to the fluctuations in the energy of a system at constant N, V and T, where A is the number of particles in a volume V at temperature T. Similarly, fluctuations in the number density (p = N/V) of an open system at constant p, V and T, where p is the chemical potential, are related to the isothemial compressibility iCp which is another response fiinction. Temperature-dependent fluctuations characterize the dynamic equilibrium of themiodynamic systems, in contrast to the equilibrium of purely mechanical bodies in which fluctuations are absent. 

The coexisting densities below are detennined by the equalities of the chemical potentials and pressures of the coexisting phases, which implies that tire horizontal line joining the coexisting vapour and liquid phases obeys the condition... 

This is Kirkwood s expression for the chemical potential. To use it, one needs the pair correlation fimction as a fimction of the coupling parameter A as well as its spatial dependence. For instance, if A is the charge on a selected ion in an electrolyte, the excess chemical potential follows from a theory that provides the dependence of g(i 2, A) on the charge and the distance r 2- This method of calculating the chemical potential is known as the Gimtelburg charging process, after Guntelburg who applied it to electrolytes. 

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