The chemical potential p, has an important function in the system's thermodynamic behavior analogous to pressure or temperature. A temperature difference between two bodies determines the tendency of heat to pass from one body to another while a pressure difference determines the tendency for bodily movement. We will show that a difference in chemical potential can be viewed as the cause for chemical reaction or for mass transfer from one phase to another. The chemical potential p, greatly facilitates the discussion of open systems, or of closed systems that undergo chemical composition changes.

10.1.2 The Gibbs Free Energy, G

Calculation of changes dU of the internal energy of a system U requires the estimation of changes of its entropy S, volume V, and number of moles . For chemical applications, including atmospheric chemistry, it is inconvenient to work with entropy and volume as independent variables. Temperature and pressure are much more useful. The study of atmospheric processes can therefore be facilitated by introducing other thermodynamic variables in addition to the internal energy U. One of the most useful is the Gibbs free energy G, defined as

Differentiating (10.11)

and combining (10.12) with (10.10), one obtains k dG= -SdT + Vdp + ^^drii (10.13)

Note that one can propose using (10.13) as an alternative definition of the chemical potential:

Both definitions (10.9) and (10.14) are equivalent. Equation (10.13) is the basis for chemical thermodynamics.

For a system at constant temperature (dT = 0) and pressure (dp = 0)

For a system under constant temperature, pressure, and with constant chemical composition (drii = 0), dG = 0, or the system has a constant Gibbs free energy. Equation (10.13) provides the means of calculating infinitesimal changes in the Gibbs free energy of the system. Let us assume that the system under discussion is enlarged m times in size, its temperature, pressure, and the relative proportions of each component remaining unchanged. Under such conditions the chemical potentials, which do not depend on the overall size of the system, remain unchanged. Let the original value of the Gibbs free energy of the system be G and the number of moles of species i, n,. After the system is enlarged m times, these quantities are now mG and mn,. The change in Gibbs free energy of the system is

and the changes in the number of moles are

k | |

(10.18) | |

i=l |

Equation (10.18) applies in general and provides additional significance to the concept of chemical potential. The Gibbs free energy of a system containing k chemical compounds can be calculated by

• + V-knk that is, by summation of the products of the chemical potentials and the number of moles of each species. Note that for a pure substance

tii and thus the chemical potential is the value of the Gibbs free energy per mole of the substance. One should note that both (10.13) and (10.18) are applicable in general. It may appear surprising that T and p do not enter explicitly in (10.18). To explore this point a little further, differentiating (10.18)

and combining with (10.13), we obtain

k | |

—SdT +Vdp = J2n' |
(10.21) |

¡=1 |

This relation, known as the Gibbs-Duhem equation, shows that when the temperature and pressure of a system change there is a corresponding change of the chemical potentials of the various compounds.

The second law of thermodynamics states that the entropy of a system in an adiabatic (dQ = 0) enclosure increases for an irreversible process and remains constant in a reversible one. This law can be expressed as dS> 0 (10.22)

Therefore a system will try to increase its entropy and when the entropy reaches its maximum value the system will be at equilibrium. One can show that for a system at constant temperature and pressure the criterion corresponding to (10.22) is dG < 0 (10.23)

or that a system will tend to decrease its Gibbs free energy. For a proof the reader is referred to Denbigh (1981).

Consider the reaction A—B, and let us assume that initially there are nA moles of A and «b moles of B. The Gibbs free energy of the system is, using (10.18)

If the system is closed nr = «a + "b = constant

and this equation can be rewritten as

where nA nA

the mole fraction of A in the system. Let us assume that the Gibbs free energy of the system is that shown in Figure 10.1. If at a given moment the system is at point K, (10.23) suggests that dG < 0 and G will tend to decrease, so xA will increase, B will be converted to A, and the system will move to the right.

If the system at a given moment is at point M, once more dG < 0, so the system will move to the left (A will be converted to B). At point L, the Gibbs free energy is at a minimum. The system cannot spontaneously move to the left or right because then the Gibbs free energy would increase, violating (10.23). If the system is forced to move, then it will return to this equilibrium state. Therefore, for a constant T and p, the point L and the corresponding composition is the equilibrium state of the system and (xA)L the corresponding mole fraction of A. At this point dG = 0. Let us consider a general chemical reaction aA + bB

which can be rewritten mathematically as aA + bB — cC — dD = 0 (10.24)

The Gibbs free energy of the system is given by

If dnA moles of A react, then, according to the stoichiometry of the reaction, they will also consume (b/a)dnA moles of B and produce (c/a)dnA moles of C and (d/a)dnA moles of D. The corresponding change of the Gibbs free energy of the system at constant T and p is, according to (10.15)

-= Ha dnA + ~ Hb dnA ~ ~ He dnA - - pD dnA a a a b c d \

At equilibrium dG — 0 and therefore the condition for equilibrium is b c d

Let us try to generalize our conclusions so far. The most general reaction can be written as

where k is the number of species, A, , participating in the reaction, and v, the corresponding stoichiometric coefficients (positive for reactants, negative for products). One can easily extend our arguments for the single reaction (10.24) to show that the general condition for equilibrium is

k | |

I>H<=() |
(10.28) |

¿=1 |

This is the most general condition of equilibrium of a single reaction and is applicable whether the reactants and products are solids, liquids, or gases.

If there are multiple reactions taking place in a system with k species k

k y, v,„ a,.=o i=i the equilibrium condition applies to each one of these reactions and therefore at equilibrium k

where vty is the stoichiometric coefficient of species i in reaction j (there are n reactions and k species).

Reactions in the H2SO4-NH3-HNO3 System Let us assume that the following reactions take place:

2NH3(g) + H2S04(g) - (NH4)2S04(s) SBfe(g) + HNOa(g) ^ N^NO^s)

Calculate the chemical potential of sulfuric acid as a function of the chemical potentials of nitric acid and the two solids.

At equilibrium the chemical potentials of gas-phase NH3, and H2SC>4 and solids (NH4)2S04 and NH4NO, satisfy

From the second equation |iniij = Pniuno, — Hhno3- Substituting this into the first, we find that

M^HjSOj — ^ PHNO, ~ UNHjnC), + ^{NKjinSOi (10.32)

Determination of the equilibrium composition of this multiphase system therefore requires determination of the chemical potentials of all species as a function of the corresponding concentrations, temperature, and pressure.

10.1.4 Chemical Potentials of Ideal Gases and Ideal-Gas Mixtures

In this section we will discuss the chemical potentials of species in the gas, aqueous, and aerosol phases. In thermodynamics it is convenient to set up model systems to which the behavior of ideal systems approximates under limiting conditions. The important models for atmospheric chemistry are the ideal gas and the ideal solution. We will define these ideal systems using the chemical potentials and then discuss other definitions.

The Single Ideal Gas We define the ideal gas as a gas whose chemical potential \i(T.p) at temperature T and pressure p is given by

where is the standard chemical potential defined at a pressure of 1 atm and therefore is a function of temperature only. R is the ideal-gas constant. Pressure p actually stands for the ratio (p/1 atm) and is dimensionless. This definition suggests that the chemical potential of an ideal gas at constant temperature increases logarithmically with its pressure. Differentiating (10.33) with respect to pressure at constant temperature, we obtain

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