Non-equilibrium Thermodynamics of Heterogeneous Systems. Signe Kjelstrup

      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].

      Chapter 4

      The entropy production for a homogeneous phase

      We derive the entropy production in a volume element in a homogeneous phase for transport of heat, mass, charge and chemical reactions. The entropy production determines the conjugate fluxes and forces in the phase. Equivalent forms of the entropy production are given.

      The second law of thermodynamics, Eq. (2.14), says that the entropy change of the system plus its surroundings is positive for irreversible processes and zero for reversible processes. The law gives the direction of a process; it does not give its rate. Non-equilibrium thermodynamics assumes that the second law remains valid locally, cf. Eq. (1.1). In this chapter, we shall derive the entropy production for a volume element in a homogeneous phase. In the following chapters, we shall find the corresponding expressions for a surface area element and a three-phase contact line element. In total, we will then be able to describe the rate of changes in heterogeneous systems.

      The change of the entropy in a

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