Читать книгу Non-equilibrium Thermodynamics of Heterogeneous Systems - Signe Kjelstrup - Страница 26

The internal energy or the various derivatives of this energy, including the partial molar energies cannot be measured, only energy differences are measurable. In order to establish a measuring scale, we introduce the standard state as a point of reference. The standard state may be chosen freely, and different conventions have been made. Common to all choices is that they can be derived from each other by well-defined measurements.

The standard state for gases used in the SI system is the state of an ideal gas at 1 bar and constant temperature. The temperature is not specified in the definition of the standard state. Standard state values are often tabulated at 298 K, however. The definition of the chemical potential then relates any state to the standard state via

(3.67)

where Vm is the molar volume. For an ideal gas, we obtain

(3.68)

where μ⁰ is the standard chemical potential and p⁰ = 1 bar. The energy of a real gas is measured with respect to this standard state, with the fugacity f replacing the pressure p of the gas. The chemical potential is as follows:

(3.69)

where the fugacity coefficient, defined by the ratio ϕ = f/p, measures the deviation of the real gas from the ideal state, see Fig. 3.5 and Exercise 3.4. The chemical potential of ideal and real gases are illustrated in this figure. When μ − μ_id < 0 and ϕ < 1, attractive forces are important.

For a liquid solution, we use a liquid state as the standard state. The numbers to be compared become then closer to each other. Two choices are common; the Raoultian standard state (used for solvents) and the Henrian standard state (used for solutes). The Raoultian standard state is the state of a solution which obeys Raoult’s law. In this law, the vapor pressure above the solution is proportional to the mole fraction of the solvent (component 1):

(3.70)

where p₁^∗ is the vapor pressure above the pure solvent. We assume that the vapor above the solution is an ideal gas. Equation (3.68) gives then:

(3.71)

Figure 3.5The chemical potential of an ideal gas (stipled line) and a real gas (whole line). The standard state, μ⁰, at p⁰ = 1 bar is shown for the ideal gas. The deviation of the chemical potential of the real gas from the corresponding ideal gas value at p is measured by the term RT lnϕ.

which can be rearranged into

(3.72)

The Raoultian standard state for the solvent is defined by

(3.73)

The Raoultian standard state is thus determined from the standard state of the ideal gas plus a term that contains the vapor pressure of the solvent. A deviation from the Raoultian state is expressed by

(3.74)

where y₁ is the activity coefficient that measures deviation from ideal behavior, and ai is the activity, a₁ = p₁/p₁^∗. The situation with y₁ > 0 is illustrated in Fig. 3.6. Such a value means that there are repulsive forces in the liquid that enhance the vapor pressure above the liquid more than in the case of a solvent that follows Raoult’s law. This standard state is used in Chapter 11 on evaporation and condensation.

Figure 3.6The phase diagram for a solution that follows Raoult’s law (p₁ = p₁^∗x₁)when x₁ → 1 and Henry’s law (p₂ = K₂m₂) when m₂ → 1.

The Henrian standard state is defined by a solution with a solute (component 2) of concentration m₂⁰ = 1 molal, that obeys Henry’s law:

(3.75)

Here, K₂ is Henry’s law’s constant. The standard state is hypothetical as no solution is known to obey Henry’s law at this concentration. The constant K₂ is determined by measuring the vapor pressure above dilute solutions. The state is determined by extrapolation of this line to m₂⁰ = 1 molal (see Fig. 3.6). By following the same procedure as above, we obtain for the chemical potential of an ideal solution:

(3.76)

Henry’s law standard state is as follows:

(3.77)

This standard state is commonly used also for electrolyte solutions. There are two particles formed in a dilute solution per formula weight of salt dissolved, leading in the ideal case to

(3.78)

or, in the non-ideal case to

(3.79)

where γ_±² is the mean square activity coefficient of the cation and anion. This standard state, or the state based on molar concentrations, μ⁰, in

(3.80)

is used with chemical potentials of electrolytes, cf. Chapter 10. The value of μ⁰ is found from μ⁺ by converting molalities into concentrations at the standard state. The mean square activity coefficient is given by Debye–Hückel’s formula, when the solution is dilute.

The chemical potential of a surface-adsorbed component can be referred to a standard state with (a hypothetical) one molal surface excess concentration of the component, or a standard state referred to unit coverage, θ, of the surface:

(3.81)

The standard state must be connected to properties of the adjacent phase, in a similar way as the ideal gas standard state is connected to the Raoultian standard state. The symbol θj = Γ_j/Γ⁰ denotes fractional coverage. Activity coefficients corrections to the ideal formula, can be substantial if the surface is polarized:

(3.82)

where ys is the surface activity coefficient of component j.

We did not specify the temperature in these definitions. In fact, any temperature can be chosen. When one wants to compare standard states at different temperatures, one will have to use the general relation which applies to any state, including the standard state:

(3.83)

Expressions for activity coefficients in gases and liquids can be found in Perry and Green [117].

Exercise 3.A.2.Express the fugacity coefficient ϕ of a gas in terms of its molar volume. The fugacity coefficient is defined as ϕ = f/p, the ratio of the gas fugacity over the pressure of an ideal gas at corresponding conditions.

•Solution: Equation (3.49) reduces to

dμ ≡ Vmdp

for a pure isothermal gas. Vm is its molar volume. This equation is true for all gases whether they are ideal or not. The fugacity of a gas is defined by (see, e.g., [100])

dμ = RT d ln f

For densities that are low enough, the gas follows the ideal gas law, pV_ideal = RT. This will always happen at some pressure p′ when p → 0

For an ideal gas, the corresponding expression is

Subtraction and introduction of p′ = 0 gives

The fugacity coefficient is expressed by measurable quantities in this formula.

¹This property of the surface has been defended by some authors [97, 98], but rejected by others [99].