Potentials chemical

The chemical potential is defined as an intensive energy function to represent the energy level of a chemical substance in terms of the partial molar quantity of free enthalpy of the substance. For open systems permeable to heat, work, and chemical substances, the chemical potential can be used more conveniently to describe the state of the systems than the usual extensive energy functions. This chapter discusses the characteristics of the chemical potential of substances in relation with various thermodynamic energy functions. In a mixture of substances the chemical potential of an individual constituent can be expressed in its unitary part and mixing part. 

For a closed system the first law of thermodynamics has defined an energy function called internal energy U, which is expressed as a function of the temperature, volume, and number of moles of the constituent substances in the system U = u(t, V, n, nc). Furthermore, the second law has defined a state property, called entropy S, of the system, which is also expressed as a function of state variables S =s(T,V,nl---nc). Thermodynamics presumes that the functions t/(r,V,n, nj and 5(7, y, I nc) exist independent of whether the system is closed or open. The energy functions of U, H, F, and G, then, apply not only to closed systems but also to open systems. 

The total differential of the internal energy U of a system can be written as a function of independent state variables such as the temperature, volume and composition of the system as shown in Eq. 5.1  

The Partial Molar Quantity of Energy and the Chemical Potential. 

We shall now choose S, V, as independent variables. The internal energy of an 

The chemical potential pj of the gaseous constituent Cj in a non-ideal gaseous mixture has the following expression  

The fugacity fj is related to the partial pressure pj by the following relationship  

Yj is the fugacity coefficient of constituent Cj in the mixture. Equation (16) becomes  

Using the following relationship between the partial pressure pj and the total pressure 

When the total pressure tends towards 0, the behaviour of the gas becomes closer and closer to ideality, and the fugacity coefficients tend towards 1. Expressions (18) and (20) can then be written as follows  

The chemical potential p is related to a measurable quantity, the acitivity a, which is also called the effective concentration  

Substituting the expanded In term into the equation of chemical potential, we have 

Now the mole fraction of component 2 can be converted to the concentration ci (in grams per milliliter) by the following approximation  

Generally, the chemical potential, p, of the (th component in an ideal solution is as follows  

R is the molar gas constant [Chapter 10, Table 10.1] T is the thermodynamic temperature Xi is the mole fraction of the ith component p° is the standard chemical potential 

The number 1000 (g kg ) in the table means 1000 g ini kg h—molality (mol kg ), c—molarity (mol L ), g—grams of solute per liter of solution (g L ), x—mole fraction of solute, iu— mass fraction of solute. Ml—molar mass of solvent (g moT ), M2—molar mass of solute (g mol ), p—density of solution (g L ). 

In a real solution, the chemical potential of the ith component is as follows  

The activity was introduced by Gilbert Newton Lewis and plays an important role in the thermodynamic description of electrolyte solutions. As can be seen, the chemical potential consists of two contribntions the standard state term and a logarithmic term. The standard state term does not depend on concentration, but the logarithmic term does. Both terms depend on the concentration scale (molality, molarity, mole fraction, etc.) because p, , x and in Equation 1.2 all depend on the concentration scale. However, the chemical potential, p does not depend on the concentration scale. The chemical potential is the most important property of a component in solntion and is a function of temperature, pressure, and composition. It would be more accurate to rewrite Equation 1.2 using additional symbols as follows  

In Equation (1.35) maximum useful work of chemical processes in a solution W depends on five parameters of state. As a matter of convenience, 

If we assume dV = 0 and dS = 0, we will have isochrone-adiabatic conditions, xmder which maximum useful work of a reversible process is equal 

Which means that under isochrone-adiabatic conditions maximum useful work is equal to change of the internal energy U. Because of this the internal energy is also called isochrone-adiabatic potential. It is, however, very difficult to provide for the constancy of volume and even more so entropy of the system. 

This sum of internal energy and work U + PV is called enthalpy and denoted usually H  

5 Free energy and chemical equilibrium 7.5.1 Chemical potential 

So far, only pure compounds have been considered, that is, systems composed of only one molecular species in one given state of aggregation. With entropy in Clausius form and expansion work only, equation 7.26 becomes  

The chemical potential is the rate of change in internal energy for a change in the number of moles of component i, when entropy, volume, and number of moles of 

In our discussion so far we have restricted ourselves to systems containing only one chemical component. We must now consider how we can treat many-component systems, in particular those systems in which the chemical composition changes, as in systems in which chemical reactions take place. For a pure substance or a system of constant chemical composition 

Let the molar fraction be Xo, = n ,/n and the partial pressure be P, = x P, so that the chemical potential of species a is given by 

6 Diffusion Behavior of Solutions without Inter-molecular Interaction 

In this section we look at several diffusion problems of dilute solutions in which no inter-molecular interactions are considered. 

The chemical potential of an ideal solution is given in the same form as that for a perfect gas (E.71)  

Let there be a dilute solution in which molecules of the solute do not collide with each other, and suppose that the molecules of the solute move under viscous drag in a homogeneous and continuous solvent. Let us consider a solution of two elements, i.e., one solute and one solvent, and the radius of each molecule of a spherical shape be Rq. Then by Stokes law of viscosity, the viscous drag is /vi = (6rr rj Ro) vi where r/ is the viscosity of the solvent, and / is the viscous coefficient. We can set this force equivalent to the negative gradient of the chemical potential of the solute  

In order to demonstrate that the compositions Xg and Xp are in equilibration, we need to introduce the chemical potential, /r, of a particular species defined as the change in free energy resulting from the addition of one atom of that species, all other species held constant. Symbolically, this is written as 

When any two systems are allowed to interact, they will always exchange matter or energy in such a way as to equalize the chemical potentials of the systems. Indeed, equilibrium is defined as equal chemical potential between the two systems. In terms of the concepts described in Chapter 2, the Fermi energy in a solid gives a measure of the ehemical potential of the lowest energy free eleetron or the highest energy free hole. 

As discussed previously, matter transport is due to the flux of atoms or vacancies driven by gradients in the concentration, which can be described by using the Pick s first law. This special case of mass transport is not applicable to those with other types of driving forces, such as gradients in pressure, electric potential, and so on. To address this issue, it is necessary to use chemical potential, instead of concentration gradients, as driving force of the diffusions. Definition and description of chemical potential can be found in various textbooks. 

For a phase with a given amount of mass and composition, at variable temperature T and pressure p, as an infinitesimal reversible process occurs, the change in the Gibbs free energy is given by  

Because the first two terms on the right-hand side of the equation are at constant mass and composition, Eq. (5.53) can be used. In this case, when a small amount of one constituent, e.g., dn moles of the Mi constituent, is introduced into the phase, with T, p, and the other n s remaining constant, the effect on the Gibbs fi-ee energy can be expressed as  

If the number of moles of a phase is increased, while T, p, and the composition are kept unchanged, Eq. (5.56) becomes  

Because the fxi is dependent only on T, p, and composition and they must be kept to be constant, Eq. (5.57) can be integrated to yield  

The free energy of an open system depends on temperature, pressure, and its composition. If the concentration of component i (number of moles m) in the system changes, the free energy of the system changes. 

This partial molar change of free energy is called the chemical potential of species i. If this species is charged (i.e., an ion) it is called the electrochemical potential of species i because the change of the free energy of the system includes a component of electrical work. 

The electrochemical potential of an electron in a (solid) phase p is equal to the Fermi potential of that phase. Because at equilibrium, dG = 0, 

This is the Gibbs equation, which is particularly important for understanding phase equilibria. A related expression, called the Gibbs-Duhem equation, states that at equilibrium the change of chemical potential of one component results in the change of the chemical potentials of all other components 

Concentration of components in a system can be expressed in different units which results in the definition of different standard states (A. 16). In a gas phase, it is customary to use partial pressure p . 

Properties such as volume, enthalpy, free energy and entropy, which depend on the quantity of substance, are called extensive properties. In contrast, properties such as temperature, density and refractive index, which are independent of the amount of material, are referred to as intensive properties. The quantity denoting the rate of increase in the magnirnde of an extensive property with increase in the number of moles of a substance added to the system at constant temperature 

Partial molar quantities are of importance in the consideration of open systems, that is those involving transference of matter as well as energy. For an open system involving two components 

Note the use of the symbol 3 to denote a partial change which, in this case, occurs under conditions of constant temperature, pressure and number of moles of solvent (denoted by the subscripts outside the brackets). 

In practical terms the partial molar volume, V, represents the change in the total volume of a large amount of solution when one additional mole of solute is added - it is the effective volume of 1 mole of solute in solution. 

Of particular interest is the partial molar free energy, G, which is also referred to as the chemical potential, jx, and is defined for component 2 in a binary system by 

Consider any thermodynamic extensive property, such as volume, free energy, entropy, energy content, etc., the value of which, for a homogeneous system, is completely determined by the state of the system, e.g., the temperature, pressure, and the amounts of the various constituents present thus, ( is a function represented by 

The derivative (dG/dni)T,p.rn.is called the partied molar property for the constituent i, and it is represented by writing a bar over the symbol for 

If the temperature and pressure of the system are maintained constant, dT and dP are zero, so that 

By general differentiation of equation (26.6), at constant temperature and pressure but varying composition, it is seen that 

Comparing this result with equation (26.5), it follows that at a given temperature and pressure 

Equation (3.12) will often be seen in another form, 

The Greek philosopher Heraclitus concluded from observations of his environment already in the fifth century before Christ that Everything flows— Nothing stands still (ndvxa pet). Creation and decay are well known in the living world but also in inanimate nature the things around us change more or less rapidly. A lot of such processes are familiar to us from everyday life (Experiment 4.1)  

Objects made of iron rust when they come into contact with air and water. Bread dries out if one leaves it at air for a longer time. 

Copper roofs turn green (so-called patina). 

Even the seemingly stable rocks ( solid as a rock ) weather. 

Experiment 4.2 Aging of acrylic acid Acrylic acid as pure substance is a wato-clear liquid strongly smelling of vinegar. If left to stand alone in a completely sealed container, it will change by itself after some time into a colorless and odorless rigid glass. 

We consider a system with different components (i) with N particles of each kind or nj = N/Na moles. G depends on the number of particles of each kind. The change dG in G may be expressed with the help of derivatives and differentials. We also include the dependence of temperature T and pressure P  

A derivative with respect to nj is called a partial molar free energy, but is better known as chemical potential (p,)  

The chemical potential is the driving force for the change in the number of molecules in a chemical process. The molecules of kind i will decrease in number spontaneously as we approach equilibrium. If G decreases when the number of molecules of kind i is decreased, Pi is positive. As is the case with temperature and pressure, and heat capacity, the chemical potential is an intensive property, which is independent of the size of the system. Extensive properties or quantity properties, on the other hand, are volume, mass, internal energy, and free energy, which depend on the amount of molecules, that is, the size of the system. 

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In thermodynamic terms the equilibrium constant is related to the standard chemical potential by the equation... 

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. 

If there are more than two subsystems in equilibrium in the large isolated system, the transfers of S, V and n. between any pair can be chosen arbitrarily so it follows that at equilibrium all the subsystems must have the same temperature, pressure and chemical potentials. The subsystems can be chosen as very small volume elements, so it is evident that the criterion of internal equilibrium within a system (asserted earlier, but without proof) is unifonnity of temperature, pressure and chemical potentials tlu-oughout. It has now been... 

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). 

The analogue of the Clapeyron equation for multicomponent systems can be derived by a complex procedure of systematically eliminating the various chemical potentials, but an alternative derivation uses the Maxwell relation (A2.1.41)... 

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. 

Note that this has resulted in the separation of pressure and composition contributions to chemical potentials in the ideal-gas mixture. Moreover, the themiodynamic fiinctions for ideal-gas mixing at constant pressure can now be obtained ... 

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)... 

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