The chemical potential of a substance

For chemical substances, chemical reactions or phase changes, the free energy, G, depends on  

If there is an amount of one substance, n-[, present in a nfixture when the amounts of aU the other substances, 3 , are kept constant, and where the temperature and pressure are also 

If the amounts of all the substances are allowed to change, but T and p are still kept constant, then the change in G will be a sum of terms for each change of amount for each substance  

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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. 

The form of the chemical potential of a substance present in very small amount in a solution was shown by Gibbs as early as 1876 to be a logarithmic function of the concentration ... 

The chemical potential of a substance can be equated to the Gibbs energy, and in this case we can write this as g per hole3 so that ... 

We see that the chemical potential of a substance must be the same in all phases in which it exists, whether this substance is considered as a component... 

For this equation to equal Equation (8.166), the chemical potential of a substance considered as a species must be identical to the chemical potentials of the same substance considered as a component. We must emphasize that the chemical potential of the component and that of the species must refer to the same mass of the substance. 

We consider only binary solutions in this discussion. The standard states of the two components are defined as the pure components, and the chemical potentials of the components are based on the molecular mass of the monomeric species. We designate the components by the subscripts 1 and 2 and the monomeric species by the subscripts A1 and B1, respectively. From the discussion given in Section 8.15 we know that the chemical potential of a substance considered in terms of the species present in a solution must be... 

Now, a question arises, Is there a way to quantitatively describe the phase boundaries in terms of P and T The phase rule predicts the existence of the phase boundaries, but does not give any clue on the shape (slope) of the boundaries. To answer the above question, we make use of the fact that at equilibrium the chemical potential of a substance is the same in all phases present. 

Obviously, the chemical potential of a substance is the partial molar quantity of the principal energy functions with respect to the number of moles of the substance at constant values of their respective independent variables in the system as shown in Fig. 5.1. 

The chemical potential of a substance i in a homogeneous mixture depends on the temperature, pressure, and concentrations of constituent substances, p, = p,(T,p,xl , ) whereas, that of a pure substance is a function of temperature and pressure only. As mentioned in the foregoing chapters, the mixing of substances causes an increase in entropy of the system and hence changes the chemical potentials of the substances... 

The chemical potential of a pure substance i indicates the thermodynamic energy level of the substance relative to the energy level of the chemical elements that make up the substance i. In chemical thermodynamics the chemical potentials of elements are conventionally all set zero in the stable state of them at the standard temperature 298 K and pressure 101.3 kPa. The chemical potential of a substance (a chemical compound) / at the standard state, as a result, is equal to the free enthalpy (Gibbs energy) required to form one mole of the substance i from its constituent elements in their stable standard state. 

A gaseous substance at dilute density normally is in the state of an ideal gas and it turns into a non-ideal gas as the density increases. A further increase in the density leads to the condensation of a gas into a liquid or solid phase. In the ideal gaseous state the chemical potential of a substance changes linearly with the logarithm of the density, and a deviation from the linearity occurs in the non-ideal state. For a condensed substance in the liquid or solid state its chemical potential hardly changes with the density. This chapter concerns the equations of state and the calculation of thermodynamic potentials of gaseous and condensed substances. 

We have made mention earlier (Frame 5, section 5.4) albeit very briefly, of the definition of the chemical potential, //, of a substance i. For a two component system having components labelled as 1 and 2, this intensive quantity (Frame 1, section 1.3) is defined as the rate of change of Gibbs energy per mole of substance present ... 

The most important chemical thermodynamic property is the chemical potential of a substance, denoted /x.18 The chemical potential is the intensive property that is the criterion for equilibrium with respect to the transfer or transformation of matter. Each component in a soil has a chemical potential that determines the relative propensity of the component to be transferred from one phase to another, or to be transformed into an entirely different chemical compound in the soil. Just as thermal energy is transferred from regions of high temperature to regions of low temperature, so matter is transferred from phases or substances of high chemical potential to phases or substances of low chemical potential. Chemical potential is measured in units of joules per mole (J mol 1) or joules per kilogram (J kg 1). 

Virtually all chemical reactions in soils are studied as isothermal, isobaric processes. It is for this reason that the measurement of the chemical potentials of soil components involves the prior designation of a set of Standard States that are characterized by selected values of T and P and specific conditions on the phases of matter. Unlike the situation for T and P, however, there is no strictly Ihermodynamic method for determining absolute values of the chemical potential of a substance. The reason for this is that p represents an intrinsic chemical property that, by its very conception, cannot be identified with a universal scale, such as the Kelvin scale for T, which exists regardless of the chemical nature of a substance having the property. Moreover, p cannot usefully be accorded a reference value of zero in the complete absence of a substance, as is the applied pressure, because there is no thermodynamic method for measuring p by virtue of the creation of matter. 

Iquaiion sl. 8 applies to any substance, in any phase, in any kind of mixture at equilibrium. It expresses the idea that the chemical potential of a substance always may be written as equal to the Standard-State chemical potential plus a... 

In summary, the chemical potential of a substance depends on its concentration, the pressure, the electrical potential, and gravity. We can compare the chemical potentials of a substance on two sides of a barrier to decide whether it is in equilibrium. If fij is the same on both sides, we would not expect a net movement of species / to occur spontaneously across the barrier. The relative values of the chemical potential of species / at various locations are used to predict the direction for spontaneous movement of that chemical substance (toward lower /a ), just as temperatures are compared to predict the direction for heat flow (toward lower T). We will also find that Afij from one region to another gives a convenient measure of the driving force on species /. 

Thermodynamic aspects of the stability of electroless plating solutions have been discussed by Vashkyalis [10]. Electroless plating solutions are thermodynamically unstable and subject to spontaneous decomposition, resulting in precipitation of metal throughout the solution. For the plating solution to be practically useful, the actual occurrence of spontaneous decomposition must be prevented. Thermodynamic conditions which must be met to prevent decomposition can be discussed on the basis of the Gibbs-Thomson equation, which relates the chemical potential of a substance to the curvature of its surface. It follows that the equilibrium potential of a metal particle... 

The chemical potential of a substance corresponds at 298 K and 100 kPa (1 bar) to its molar Gibbs energy of formation if the zero points of the potential scales are appropriately chosen. Therefore, when looking for the chemical potential, tables in which this quantity is listed can be used. 

For liquid solutions, we need to express how the chemical potential of a solution va ries with its composition. In order to derive a useful expression, we should remember that, in equilibrium, the chemical potential of a substance in the liquid phase must be equal... 

The increase in the chemical potential of a substance in the dispersed state is formally related to the surface curvature of the particles but in fact, according to eq. (1.11), the chemical potential increase is due to the increase in the surface area fraction (and therefore in the surface free energy) per unit volume with decreasing particle size. 

Ordinarily for purposes of illustration we shall use molar concentrations in the equilibrium constant, realizing that to be precise we should use the activities. One misunderstanding that arises because of this replacement of activity by concentration should be avoided. The activity is sometimes regarded as if it were an effective concentration. This is a legitimate formal point of view however, it is deceptive in that it conveys the incorrect notion that activity is designed to measure the concentration of a substance in a mixture. The activity is designed for one purpose only, namely to provide a convenient measure of the chemical potential of a substance in a mixture. The connection between activity and concentration in dilute solutions is not that one is a measure of the other, but that either one is a measure of the chemical potential of the substance. It would be better to think of the concentration in an ideal solution as being the effective activity. 

N appears two times oti the left as well as on the right side of the conversion formula H, however, appears six times. Therefore, if the chemical potential of a substance is increased by a fixed, although arbitrary summand (say 1,000 kG, as shown above in the third line) for every H appearing in its content formula, this added value cancels when we compute the potential difference and we end up with the same result as in the second line above. The same holds for nitrogen. This means that the reference level for any element could in principle be chosen arbitrarily as mentioned earlier. For the sake of simplicity, the value 0 is assigned to the chemical potential of all elements. 

The chemical drive A T), which is calculated from the temperature-dependent potentials, exhibits a noticeably more linear gradient than the p T) curves. Both curves intersect at the standard temperature T because the chemical potential of a substance at standard conditions corresponds to the drive to decompose into the elements (here sodium and chlorine). 

Fig. 5.4 Temperature dependence of the chemical potentials of a substance as solid, melt, or vapor.

Pressure Coefficient As previously stated, the value of the chemical potential of a substance depends not only upon temperature but upon pressure as well. Moreover, the potential generally increases when the pressure increases (Fig. 5.7). 

Fig. 5.10 Temperature dependence of the chemical potentials of a substance in solid, liquid, and gaseous states at low pressure lower curves) and at high pressure (upper curves) (in case of 0<,5,

If one is interested, for example, in the temperamre coefficient of the chemical potential of a substance, it is only necessary to find the value of the corresponding molar entropy in an appropriate table book and to change the sign. 

Charge number (of a type i of particles, cations, anions) (16, 535) Temperature coefficient of the chemical potential (of a substance B) (131)... 

Chemical potential The chemical potential of a substance is the extent to which the free energy of a system changes in response to a change in the amount of that substance. 

Thus, in so far as wo need to allow for the effect of the gravitational field, the chemical potential of a substance is not equal throughout the depth of a phase, but it is the sum of the terms /C and Mtgh which has this property of constancy. 

We begin by obtaining the chemical potential of a solution. The general expression for the chemical potential of a substance is T) = p°(po5 T)+... 

Because of the importance of the Gibbs energy in spontaneity considerations, the majority of derivatives with respect to n concern G. The chemical potential of a substance, /x, is defined as the change in the Gibbs energy with respect to amount at constant temperature and pressure ... 

We need an explicit formula for the variation of the chemical potential of a substance with the composition of the mixture. Here we use the strategy mentioned at the start of the chapter we begin by considering the chemical potential of a gas, not because gases are particularly interesting in biology but because we can use the resulting expression to derive results for solutions. 

Both the chemical potential of a substance i, jUf, and the potential at a point are quantities of energy. jXi is the change in Gibbs free energy of the system when an infinitesimal amount of material i is transferred into it from the standard state of i (keeping the temperature, pressure and concentrations of the other components constant) the potential, on the other hand, is the... 

A measure of the chemical potential of a substance, where chemical potential is not equal to concentration, that allows mathematical relations equivalent to those for ideal systems to he used to correlate changes in an experimentally measured quantity with changes in chemical potential. 

The chemical potential of two phases at equilibrium must be identical, and so the chemical potential of a substance as a function of temperature is continuous at the phase transition. The discontinuous functions just mentioned, volume, entropy, enthalpy, and heat capacity, are all related to first derivatives of the chemical potential. Therefore, the type of transitions we have considered so far, the ones to which we are most accustomed, is categorized as having a discontinuity in the first derivative of the chemical potenhal with respect to temperature at the phase transition. These are called first-order phase transitions. 

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