Free-energies

Free energy denoted by the letter G is a thermodynamic quantity, with which we can predict the spontaneity of a reaction. The change in free energy (AG) is related to the changes in enthalpy and entropy, and temperature as indicated by the equation  

AG represents the change in free energy, AH represents the change in enthalpy, AS represents the change in entropy, and T represents the temperature. 

Knowing the energy function H o) allows us, at least in principle, to calculate the average (or expected) value of any observable O of the system  

Using equation 7.3, U can also be expressed in terms of the free energy F  

The basic problem of statistical mechanics is to evaluate the sum-over-states in equation 7.2 and obtain Z and F as functions of T and any other variables (such as external magnetic fields) that might appear in %. Any thermodynamic observable of interest can then be obtained in a straightforward manner from equation 7.5. In practice, however, the sum-over-states often turns out to be prohibitively difficult to evaluate. Instead, the physical system is usually replaced with a simpler model system and/or some simplifying approximations are made so that the sum-over-states can be evaluated directly. 

The free energy is derived from the entropy and is, in many ways, a more useful function to use. The free energy which is referred to when we are discussing processes at constant pressure is the Gibbs free energy (G). This is defined by 

The change in the free energy at constant temperature arises from changes in enthalpy and entropy and is 

This nonexpansion work can be extracted from the system as electrical work, as in the case of a chemical reaction taking place in an electrochemical cell, or the energy can be stored in biological molecules such as adenosine triphosphate (ATP). 

When the system has attained an equilibrium state it no longer has the ability to reverse itself. Consequently all spontaneous processes are irreversible. The fact that all spontaneous processes taking place at constant temperature and pressure are accompanied by a negative free energy change provides a useful criterion of the spontaneity of any given process. 

By applying these concepts to chemical equilibria we can derive (see Box 3.2) the following simple relationship between free energy change and the equilibrium constant of a reversible reaction, K  

Gibbs free energy, G = H -TS, or Helmholtz free energy, F = U - TS, are the key quantities in the description of chemical equilibrium. For a given chemical process, consider the following equations  

The Gibbs free energy, symbol G, is defined by the following equations  

Equations 7.33-7.34, the conditions for spontaneous evolution, when written in terms of G at constant T and P, lead to the following expression  

This is the fundamental equation of chemical equilibrium. It applies to any chemical system (not just isolated systems) at constant temperature and pressure, that is, the normal working conditions of chemistry. To understand how it works, consider the simple example of a transformation of a pure substance between states A and B, with d A = -d B equation 7.49 says that ( Ta - Tb) d A 0, so if Ta l B then a must decrease - hence p, as chemical potential. At equilibrium, Ta = ItB so that the sign of d A is irrelevant the system has no driving force to evolve either way. 

The Helmholtz free energy, A = U - TS, is an equivalent form, more useful in conditions of constant temperature and volume and hence of lesser use in practical thermodynamics. Its application will be discussed in Section 9.7. 

Biological systems do not consume energy they transform it. Energy available as food for animals, sunlight for plants, and energy-rich compounds for microbes is transformed from whatever form it is at the source into a form more useful to the living system. There is always waste (up to 90% or more) in these transformations. 

We know, however, that many energy transformations never happen spontaneously. Cold objects don t heat themselves, blood does not rush throughout the veins and arteries without a heart to pump it, and dust on the floor does not self-organize into a beautiful plant. Some things just don t happen by themselves. 

Why is this so It was explained in a previous section on effort and flow variables (Section 2.1) that flow always occurs from points of higher effort to lower, never the other way around. Perhaps that explanation is sufficient. We usually have enough experience with real effort variables (pressure, gravity, temperature, etc.) to expect that the impossible just won t happen. And it doesn t. 

Thermodynamics is the science of energy transformations, and thermodynamicists have a different method they use to explain the spontaneity of energy transformations. They call it free energy, and the sign of free energy determines whether or not the transformation will proceed the amount of free energy determines whether or not the transformation will take place with gusto. 

There are several definitions to be presented, and these can be quite confusing the first time they are encountered. This is natural, and should not be taken as an excuse to forget this whole approach. With time and acceptance, you will probably find yourself at ease with the free energy approach. 

For the three systems discussed here, the free energy P is split into a hard-sphere part Fo and an attractive part Fa. It is most convenient to consider the normalized free energy/= FvjV, where v = 4jta /3 is the volume of a (colloidal) particle and Vis the volume of the system. We thus write  

The hard-sphere part depends only on the (colloid) volume fraction T and is the same in the three cases. We have to differentiate between a fluid (F) phase [either gas (G) or liquid (L)] and a crystalline solid (S). Expressions are available in the 

For the attractive part j]] we need the specifics of the system, as discussed in the next sections. For all three systems we consider first the pair potential W H), where H is the interparticle distance H is zero for particles in contact We shall use only the relative particle separation h = HI a, where a is the particle radius. In all cases W(h) = for h 0. The pair potential is characterized by the strength e = — W(0) and by the relative range q = KT ja, where is the range of the attraction. From an appropriate expression for W h) as a function of e and q the attractive free energy can be derived, and from f=fo+fii as a function of T], e and q all thermodynamic properties (including the phase behavior) follow from standard thermodynamics. For example, the chemical potential i of the spherical particles and the pressure p of the system are given by  

When the attractive contribution to / p, and pv is zero, we have a pure hard-sphere system where the only possible phase coexistence is FSatq = 0.492 andq = 0.542, according to Equations 7.2 and 7.3, with results nearly identical to that of computer simulations [5]. Atthis coexistence p[J = 15.463 kT and (pv) = 6.081 kT. When there is an attractive component the phase behavior is richer, with the possibility of two-phase GL, GS and LS coexistence and a GLS triple point, just like in simple atomic or molecular systems. 

Once expressions for p andpv (and the first and second derivatives p and p with respect to q) have been found, the complete phase diagram for each system may be calculated. Binodal points and triple points follow from equal p and pv in two or three phases, respectively. Critical GL points are obtained from p = p = 0. Of central importance for the existence of a stable liquid is the critical endpoint (cep) its 

The second key equation having to do with equilibrium relates to the difference in energy between reactants and products. The particular form of energy important in this relationship is free energy, G. The difference in free energy between product and reactant states is AG = Gp Q y j - G, The relationship between free energy and equilibrium is 

In these equations, R is the gas constant (0.08206 L atm/(mol K) or 8.314 L kPa/(mol K), depending on your units see Chapter 11 for more information), Tis temperature, and In refers to the natural logarithm (log base e). The equation is typically true for reactions that occur with no change in temperature or pressure. 

Favorable reactions possess negative values for AG, and unfavorable reactions possess positive values for AG. Energy must be added to drive an unfavorable reaction forwcird. If the AG for a set of reaction conditions is 0, the reaction is at equilibrium. 

After waiting three hours, you measure the following concentrations  

You may have raced through these problems, or you may have moved at the speed of a dead snail in winter. But rate is separate from equilibrium — whatever your pace, you ve made it this far. Now shift in opposition to any perturbing problems check your work. 

In thermodynamics, entropy enjoys the status as an infallible criterion of spontaneity. The concept of entropy could be used to determine whether or not a given process would take place spontaneously. It has been found that in a natural or spontaneous process there would be an increase in the entropy of the system. This is the most general criterion of spontaneity that thermodynamics offers however, to use this concept one must consider the entropy change in a process under the condition of constant volume and internal energy. Though infallible, entropy is thus not a very convenient criterion. There have, therefore, been attempts to find more suitable thermodynamic functions that would be of greater practical 

Helmholtz free energy, represented by the symbol A, is defined as  

In this expression U is the internal energy, T is the absolute temperature and S is the entropy. 

Gibbs free energy, represented by the symbol G (the symbol F is also sometimes used in place of symbol G), is defined as  

Keeping in view the expression for G, one obtains, in the differential form, 

In electrical and electrochemical processes, electrical work is defined as the product of charges moved (Q) times the potential (E) through which it is moved. If this work is done in an electrochemical cell in which the potential difference between its two half-cells is E, and the charge is that of 1 mol of reactant in which n mol of electrons are transferred, then the electrical work (w) done by the cell must be - E. In this relationship, the Faraday constant F is required to convert coulombs from moles of electrons. In an electrochemical cell at equilibrium, no current flows and the energy change occurring in a reaction is expressed in Eq. (4.1). 

Under standard condition, the standard free energy of the cell reaction AG° is directly related to the standard potential difference 

For solids, liquid compoimds, or elements, standard condition is the pure compoimd or element for gases it is 100 kPa pressure and for solutes it is the ideal 1M (mol/L) concentration. 

Electrode potentials can be combined algebraically to give cell potential. For a galvanic cell, such as the Daniell cell shown previously in Chap. 3, a positive cell voltage will be obtained if the difference is taken in the way described in Eq. (4.3) and illustrated in Fig. 4.1. 

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TABLE 10.4 Interplay of ASsys and ASsurr in Determining the Sign of ASunjV 

Signs of Entropy Changes ASSyS ASsurr ASuniv 

No (process will occur in opposite direction) Yes, if ASsys has a larger magnitude than ASsurr Yes, if ASsurr has a larger magnitude than ASsys 

We can now understand why spontaneity is often dependent on temperature and thus why water spontaneously freezes below 0°C and melts above 0°C. The term ASsurr is temperature-dependent. Since 

The symbol G for free energy honors Josiah Willard Gibbs (1839-1903), who was Professor of Mathematical Physics at Yale University from 1871 to 1903. He laid the foundations of many areas of thermodynamics, particularly as they apply to chemistry. 

Click Coached Problems for a self-study module on putting Aff and AS together. 

A spontaneous process is capable of producing useful work. 

The sign of the free energy change can be used to determine the spontaneity of a reaction carried out at constant temperature and pressure. 

If AG is positive, the reaction will not take place spontaneously. Instead, the reverse reaction will be spontaneous. 

It is important to realize that this is the total entropy change, because mixing could result in heat being emitted to, or absorbed from, the surroundings, which would change its entropy. We can therefore write the total entropy change as in Equation 10-18. 

26 There are various approaches, in addition to Boltzmann s equation, that can be taken to calculate thermodynamic parameters (e.g., through the use of the paitition function). 

We can now define the free energy for a constant pressure process (i.e., the Gibbs free energy) to be as in Equation 10-21. 

It takes work to organize chemical molecules an increase in the level of organization is reflected in a decrease in the entropy term (more organization, less uniformity). Conversely, a disorganized, more random system of mole- 

The change in Gibbs free energy (AG), which occurs as a system proceeds toward equilibrium, can be expressed as the sum of two terms. The first term is the standard free energy change (A G°), which is fixed for any given reaction. AG° can be calculated from the stoichiometry of the reaction (i.e., how many moles of one compound react with how many moles of another compound) and the standard free energies of the chemicals involved. The second term contains the reaction quotient (Q), which depends on the concentrations of chemicals present. The fact that AG can be expressed in terms of the concentrations of all chemicals present in a system makes it possible to determine in which direction a chemical reaction will proceed and to predict its final composition when it reaches equilibrium. 

To get a better feel for Gibbs free energy, consider a reversible reaction occurring in water, where lowercase letters represent stoichiometric coefficients and uppercase letters represent four chemical compounds, A, B, C, and D  

As this system proceeds toward equilibrium, the change in Gibbs free energy per additional mole reacting is 

When a substance absorbs a quantity of heat dq, its temperature will rise accordingly by an amount dT. The ratio of the two is the heat capacity, defined as 

Since dq is not a state function, c will depend on the path. The problem can be simplified by introducing the enthalpy function, defined as 

If the heat capacity measurement is carried out at constant pressure dP — 0, and since by definition dw = —PdV, it follows from Eq. (5.3) that dH = dq p. In other words, the heat absorbed or released by any substance at constant pressure is a measure of its enthalpy. 

From this result it follows from Eq. (5.1) that 

Given that there is no absolute scale for energy, the best that can be done is to arbitrarily define a standard state and relate all other changes to that state — thermodynamics deals only with relative changes. Consequently and by convention, the formation enthalpy of the elements in their standard state at 298 K is assumed to be zero i.e., for any element. 

An important example of this is the formation of H in several bacterial fermentation processes which is exergonic only under low H partial 

Note that is positive, as it should be, since this reaction is exothermic and heat flow occurs to the surroundings, increasing the randomness of the surroundings. 

In this case AXurr is negative because heat flow occurs from the surroundings to the system. 

We have seen that the spontaneity of a process is determined by the entropy change it produces in the universe. We also have seen that ASu iv has two components, AS y and 

AXurr- If for some process both AXys and AS u re positive, then AS , the process is spontaneous. If, on the other hand, both AXys and AXun process does not occur in the direction indicated but is spontaneous in the opposite direction. Finally, if AS y and AXum have opposite signs, the spontaneity of the process depends on the sizes of the opposing terms. These cases are summarized in Table 17.3. 

Interplay of A5sys and ASjurr in Determining the Sign of ASuniv Signs of Entropy Changes 

The result of Ag(S) calculation is shown in Fig. 6.6. Considering a cooling process from the stable isotropic phase we shall better understand the physical sense of the three critical temperatures. For T T,A (dot curve 3 in the figure) the absolute minimum is situated at 5 = 0 and this corresponds to the stable isotropic phase. As the temperature approaches Tc from above, in the range of Tc T Tc, a second minimum appears above the abscissa axis, which corresponds to the 

If is 0, the system is at equilibrium there is no tendency for reaction to occur in either direction. 

In other words, AG is a measure of the driving force of a reaction. Reactions, at constant pressure and temperature, go in such a direction as to decrease the free energy 

Experimentally, it is found that for polymer solutions, the entropy of mixing is substantially overestimated by the expression in Equation 2.3. To correct this problem, Xh is replaced by an interaction parameter (x) that comprises an entropic component independent of temperature and denoted by P and the enthalpic component (Xh) such that a practical expression for AG is then 

which is sometimes referred to as the lattice parameter, is usually around 0.35 0.1 

Furthermore, in comparison with experiments, it turns out that % is also a function of concentration. Most listed values of % are for the particular polymer in the given solvent at a stated temperature and at infinite dilution. Some values of x are listed in Table 2.3 for polymers in what are known as good solvents. A number of more rigorous expressions for and AG have been presented over the years [15]. The predictions of various theories have been compared with Monte Carlo simulations [16,17] that are able to compute thermodynamic properties with fewer assumptions than in the original models. 

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CALCULATE RESIDUAL CONTRIBUTION TO EXCESS FREE ENERGY lAO DO 141 l =lfN... 

Gibbs function (see Gibbs free energy, free energy) gram... 

Gibbs-Helmholtz equation This equation relates the heats and free energy changes which occur during a chemical reaction. For a reaction carried out at constant pressure... 

Helmholtz free energy The maximum amount of energy available to do work resulting from changes in a system at constant volume. See free energy and Gibbs-Helmholtz equation. 

A quantitative theory of rate processes has been developed on the assumption that the activated state has a characteristic enthalpy, entropy and free energy the concentration of activated molecules may thus be calculated using statistical mechanical methods. Whilst the theory gives a very plausible treatment of very many rate processes, it suffers from the difficulty of calculating the thermodynamic properties of the transition state. 

The relations which permit us to express equilibria utilize the Gibbs free energy, to which we will give the symbol G and which will be called simply free energy for the rest of this chapter. This thermodynamic quantity is expressed as a function of enthalpy and entropy. This is not to be confused with the Helmholtz free energy which we will note sF (L" j (j, > )... 

G = Gibbs molar free energy S = molar entropy F = Helmholtz free molar energy H = molar enthalpy U = molar internal energy... 

At the triple point, the free energies of each phase are equal ... 

Gy = molar free energy of the liquid G = molar free energy of the gas... 

The determination of equilibria is done theoretically via the calculation of free energies. In practice, the concept of fugacity is used for which the unit of measurement is the bar. The equation linking the fugacity to the free energy is written as follows >... 

Gg = partial free energy of component i in the gas phase at temperature T and pressure P [kJ/kmol]... 

G = free energy of the phase considered at the temperature Tand pressure P... 

Gibbs free energy or Gibbs molar free energy molar flow of gas phase acceleration of gravity enthalpy, molar enthalpy, weight enthalpy Henry s constant Planck s constant height horsepower radiation intensity molar flux... 

A general prerequisite for the existence of a stable interface between two phases is that the free energy of formation of the interface be positive were it negative or zero, fluctuations would lead to complete dispersion of one phase in another. As implied, thermodynamics constitutes an important discipline within the general subject. It is one in which surface area joins the usual extensive quantities of mass and volume and in which surface tension and surface composition join the usual intensive quantities of pressure, temperature, and bulk composition. The thermodynamic functions of free energy, enthalpy and entropy can be defined for an interface as well as for a bulk portion of matter. Chapters II and ni are based on a rich history of thermodynamic studies of the liquid interface. The phase behavior of liquid films enters in Chapter IV, and the electrical potential and charge are added as thermodynamic variables in Chapter V. 

AP such that the work against this pressure difference APAvr dr is just equal to the decrease in surface free energy. Thus... 

This is exact—see Problem 11-8. Notice that Eq. 11-14 is exactly what one would write, assuming the meniscus to be hanging from the wall of the capillary and its weight to be supported by the vertical component of the surface tension, 7 cos 6, multiplied by the circumference of the capillary cross section, 2ar. Thus, once again, the mathematical identity of the concepts of surface tension and surface free energy is observed. 

Figure III-l depicts a hypothetical system consisting of some liquid that fills a box having a sliding cover the material of the cover is such that the interfacial tension between it and the liquid is zero. If the cover is slid back so as to uncover an amount of surface dJl, the work required to do so will he ydSl. This is reversible work at constant pressure and temperature and thus gives the increase in free energy of the system (see Section XVII-12 for a more detailed discussion of the thermodynamics of surfaces).

The total free energy of the system is then made up of the molar free energy times the total number of moles of the liquid plus G, the surface free energy per unit area, times the total surface area. Thus... 

The total surface energy generally is larger than the surface free energy. It is frequently the more informative of the two quantities, or at least it is more easily related to molecular models. 

A very important thermodynamic relationship is that giving the effect of surface curvature on the molar free energy of a substance. This is perhaps best understood in terms of the pressure drop AP across an interface, as given by Young and Laplace in Eq. II-7. From thermodynamics, the effect of a change in mechanical pressure at constant temperature on the molar h ee energy of a substance is... 

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