Fugacity, Activity and Chemical Potential

1. Fugacity, Activity and Chemical Potential

2. Chemical Potential Chemical potential μis partial molar free energy. For a gaseous species i at partial pressure Pi and constant temperature: μi = μi° + RT lnPi where μi°is the chemical potential in a standard state, which is a function of temperature only. This relationship holds only if i is an ideal gas; it obeys the gas law. It is possible to correct for a departure from ideality by means of an equation of the form: μi = μi° + RT lnPi + A(Pi-1) + B(Pi2 -1)/2 + . . . where A and B and further terms are empirical, experimentally determined constants.

3. Lewis (1907) introduced the concept of fugacity f, such that, μ = μ° + RT ln(f/f°) where f° is the fugacity in a standard state. The standard state is often taken as 1 atm and T, but it is possible to select any con- venient standard state. Note: By convention, μi* is a function of T and P, whereas μi°, as used here, is a function of T only. Lewis derived fugacity by the consideration of: _ V = RT/P for an ideal gas, whereas, _ V = RT/f for a real gas. _ Then dG = (RT/f)df = VdP _ or RTd lnf = VdP This shows that P-V-T data are needed in order to evaluate f.

4. Fugacity has the property lim f/P = 1 P→0 but this is not a definition of fugacity. f/P is defined as the fugacity coefficient γ f/P = γ, so f = γP Lewis also defined activitya as the fugacity ratio: f/f° = a Since a is a ratio, it is always unitless regardless of what units were involved in the fugacity.

5. If we differentiate μ with respect to pressure, keeping in mind that μ° is a function of T, (δμ/δP)T = RT(δln a/δP)T but we can also determine that from dG = SdT – PdV _ so, RT dln a = VdP If we subtract RTdln P from both sides of the equation, _ RTdln a – RTdlnP = VdP – RTdlnP rearranging RTdln(a/P) = (V – [RT/P])dP Rearranging and integrating between 0 and P,

6. _ If V as a function of P, V(P), is replaced by an equation of state, adjustment for non-ideality can be made. (An equation of state is one that relates the properties of P, T and V; a number of_ these have been developed, e.g. Redlich-Kwong.) Note that if V(P)= RT/P, the gas is ideal and the value of the integral is zero. However, an activity coefficient γ = a/P is defined by:

7. So far, we have dealt with gases only, but the concept of activity allows us to expand to solutions and solids. In solutions, mmolality = moles solute per 1000g of water ( At 4°C, 1000g of water is 1 litre.) Molality gives us both the number of moles of solute and solvent and is what is used in thermodynamics. In solutions, there are some complications. Commonly, we can say ai = γimi, where ai → γimi as mi → 0 or a standard state for solutions in which γi° = 1 for mi° at infinite dilution. There are problems with this in its application, however. Without elaboration, the only standard state that works without complications is mi° = 1. In the case of solids, e.g. solid solutions, the situation is easier because we can define the standard state as the pure solid. Then Xi the mole fraction works well, although we still need to evaluate an activity coefficient γi.

8. In looking at solutions in general, i.e. both liquid and solid solutions, the simplest relationship is Raoult’s law: ai = Xi for component i (the solute) in a solution However, real solutions generally depart significantly from ideal, or Raout’s law, solutions. A first order sort of correction is Henry’s law: ai = kXi ,where k is the Henry’s law coefficient. As can be seen from the diagram below, Henry’s law applies in dilute solutions fairly well, and Raoult’s law can often be seen to be a good approximation for concentrated solutions.