## PT dp J T dp

But is not a function of pressure and therefore its derivative with respect to pressure is zero, so for an ideal gas

Using (10.13), one can calculate the derivative of G with respect to pressure

Dr-V 0036, but for a system consisting of n moles of a single gas [see (10.18)]

and therefore from (10.36) and (10.37)

Combining (10.35) and (10.38), we see that our definition of an ideal gas entails pV = nRT

Deviations from ideal gas behavior are customarily expressed in terms of the compressibility factor C = pV/(nRT). Both dry air and water vapor have compressibility factors C, in the range 0.998 < C < 1 for the pressure and temperature ranges of atmospheric interest (Harrison 1965). Hence both dry air and water vapor can be treated as ideal gases with an error of less than 0.2% for all the conditions of atmospheric interest.

The Ideal-Gas Mixture A gaseous mixture is defined as ideal if the chemical potential of its ith component satisfies p, == \i;(T) + RT \np + RT In v, (10.40)

where p°(r) is the standard chemical potential of species i, p is the total pressure of the mixture, and y, is the gas mole fraction of compound i. For y, = 1 (pure component), (10.40) is simplified to (10.33) and therefore is precisely the same for both equations. This standard chemical potential is the Gibbs free energy per mole (recall for pure compounds G = pn) of the gas in the pure state and pressure of 1 atm. By defining the partial pressure of compound i as

 Pi = yiP (10.41) a more compact form of (10.40) is ft = tf (r) + RT In pi (10.42)

One can also prove that (10.42) is equivalent to the more traditional ideal-gas mixture definition:

PiV = riiRT

The atmosphere can be treated as an ideal gas mixture with negligible error.1

### 10.1.5 Chemical Potential of Solutions

Atmospheric aerosols at high relative humidities are aqueous solutions of species such as ammonium, nitrate, sulfate, chloride, and sodium. Cloud droplets, rain, and so on are also aqueous solutions of a variety of chemical compounds.

'For a discussion of non-ideal-gas mixtures, the reader is referred to Denbigh (1981). Discussion of these mixtures is not necessary here, as the behavior of all gases in the atmosphere can be considered ideal for all practical purposes.

Ideal Solutions A solution is defined as ideal if the chemical potential of every component is a linear function of the logarithm of its aqueous mole fraction x,-, according to the relation p, = p*(7»+flrinx, (10.43)

A multicomponent solution is ideal only if (10.43) is satisfied by every component. A solution, in general, approaches ideality as it becomes more and more dilute in all but one component (the solvent). The standard chemical potential p* is the chemical potential of pure species /(x,- = 1) at the same temperature and pressure as the solution under discussion. Note that in general p* is a function of both T and p but does not depend on the chemical composition of the solution.

Let us discuss the relationship of the preceding definition with Henry's and Raoult's laws, which are often used to define ideal solutions. Assuming that an ideal solution of i is in equilibrium with an ideal gas mixture, we have

and at equilibrium

The standard chemical potentials p* and p° are functions only of temperature and pressure, and therefore the constant A", is independent of the solution's composition.

If x, = 1 in (10.44), then Ki(T,p) is equal to the vapor pressure of the pure component i, p°, and the equation can be rewritten as

Equation (10.45) states that the vapor pressure of a gas over a solution is equal to the product of the pure component vapor pressure and its mole fraction in the solution. The lower the mole fraction in the solution, the more the vapor pressure of the gas over the solution drops. Thus (10.45) is the same as Raoult's law.

Most solutions of practical interest satisfy (10.43) only in certain chemical composition ranges and not in others. Let us focus on a binary solution of A and B. If the solution is

FIGURE 10.2 Equilibrium partial pressures of the components of an ideal binary mixture as a function of the mole fraction of A, xA.

FIGURE 10.2 Equilibrium partial pressures of the components of an ideal binary mixture as a function of the mole fraction of A, xA.

ideal for every composition, then the partial pressures of A and B will vary linearly with the mole fraction of B (Figure 10.2). When xA = 0, the mixture consists of pure B and the equilibrium partial pressure of B over the solution is pg and of A zero. The opposite is true at the other end, where xA = 1 and p,\ = pA-

The partial pressures of A and B in a realistic mixture are shown in Figure 10.3. Note that the relationships between pA,pB, and xA are nonlinear with the exceptions of the limits of xA —> 0 and xA —> 1. When xA —► 1, we have a dilute solution of B in A. In this regime

and Raoult's law applies to A. In the same regime

where H'R is a constant calculated from the slope of the pB line as xA —> 1. This relationship corresponds to Henry's law, and H'B is the Henry's law constant2 (based on

2 For a dilute aqueous solution (10.47) is equivalent to

The Henry's law constant HB defined in Chapter 7 is then related to H'B by

Partial Pressure

FIGURE 10.3 Equilibrium partial pressures of the components of a nonideal mixture of A and B. Dashed lines correspond to ideal behavior.

FIGURE 10.3 Equilibrium partial pressures of the components of a nonideal mixture of A and B. Dashed lines correspond to ideal behavior.

mole fraction) for B in A (equal to the slope of the line BN). At the other end (xA —> 0) we have and B obeys Raoult's law while A obeys Henry's law.

Summarizing, if a solution is ideal over the whole composition range (often called "perfect" solution), (10.44) is satisfied for every x,. In this case Kj is equal to the vapor pressure of pure i, which is also equal in this case to the Henry's law constant of i. Nonideal solutions approach ideality when the concentrations of all components but one approach zero. In that case the solutes satisfy Henry's law (/?,■ = H\x{), where the solvent satisfies Raoult's law (pj = p°Xj).

Nonideal Solutions Atmospheric aerosols are usually concentrated aqueous solutions that deviate significantly from ideality. This deviation from ideality is usually described by introducing the activity coefficient, y,, and the chemical potential is given by

The activity coefficient y; is in general a function of pressure and temperature together with the mole fractions of all substances in solution. For an ideal solution y(- = 1. The

standard chemical potential p* is defined as the chemical potential at the hypothetical state for which 7, —> 1 and x, —* 1 (Denbigh 1981). The product of the mole fraction x, of a solution component and its activity coefficient y(- is defined as the activity, a,, of the component

 a,- = y inl (10.50) and the chemical potential of a species i is then given by p,. = p*(2»+flr In a, (10.51)

For convenience, the amount of a species in solution is often expressed as a molality rather than as a mole fraction. The molality of a solute is its amount in moles per kilogram of solvent. For an aqueous solution containing «¡- moles of solute and nw moles of water (molecular weight 0.018 kg mol ') the molality, m„ of the solute is

Another measure of solution concentration is the molarity, expressed in moles of solute per liter of solvent (denoted by M). For water solutions at ambient conditions, because 1 liter weighs 1 kilogram, the molality and molarity of a solution are practically equal.

Traditionally, the activity of the solvent is almost always defined on the mole fraction scale ((10.40) and (10.50)), but the activity coefficient of the solute is often expressed on the molality scale:

In this case p? is the value of the chemical potential as m, —> 1 and y; —> 1.

The activity coefficients y, are determined experimentally by a series of methods including vapor pressure, freezing-point depression, osmotic pressure, and solubility measurements (Denbigh 1981).

Pure Solid Compounds The chemical potential of a pure solid compound i can be easily derived from (10.43) by setting x, — 1, so that

The chemical potential of the solid is therefore equal to its standard potential and is a function only of temperature and pressure.

Solutions of Electrolytes Most of the inorganic aerosol components dissociate on dissolution; for example, NH4NO3 dissociates forming NH| and NO J. The concentration of each ion in the aqueous solution is traditionally expressed on the molality scale and the chemical potential of each ion in a NH4NO3 solution is hnh4+ = ln(ynh+mnh4+)

where wNH+ and m^o, are the ion molalities and yNH+ and yNO- the corresponding activity coefficients. The dissociation reaction