ACTIVITY AND ACTIVITY COEFFICIENT Chapter 6 ACTIVITY AND ACTIVITY COEFFICIENT

IDEAL SOLUTION The ideal gas serves as a standard to which real gas behavior can be compared Residual properties The ideal solution serves as a standard to which real-solution behavior can be compared Excess properties

Equation (4.43) establishes the behavior of species i in an ideal-gas mixture: This equation takes on a new dimension when Gig, the Gibbs energy of pure species i in the ideal-gas state, is replaced by Gi, the Gibbs energy of pure species i as it actually exists at the mixture T and P and in the same physical state (real gas, liquid, or solid) as the mixture. It then applies to species in real solutions. We therefore define an ideal solution as one for which: (6.1)

All other thermodynamic properties for an ideal solution follow from Eq. (6.1). (4.13) (6.2) (6.3)

All other thermodynamic properties for an ideal solution follow from Eq. (6.1). The partial entropy results from differentiation with respect to temperature at constant pressure and composition and then combination with the eqs. (6.2) and (6.3) written for an ideal solution: (6.4) or (6.5)

Similarly: (6.6) (6.7) Since substitutions by Eqs. (6.1) and (6.5) yield: or (6.8)

The summability relation applied to the special case of an ideal solution is written: Application to Eqs. (6.1) through (6.8) yields: (6.9) (6.10) (6.11) (6.12)

The lewis/randall rule Equation (5.1) for ideal gas: (5.1) For ideal solution: (6.13) Substituting eq. (6.1) to eq. (6.13) yields: (6.14)

Gi is defined in eq. (4.46): (4.46) Substituting eq. (4.46) to eq. (6.14) yields: (6.15) This equation, known as the Lewis/Randall rule, applies to each species in an ideal solution at all conditions of temperature, pressure, and composition.

Division of both sides of Eq. (6.15) by P xi or (6.16) Thus the fugacity coefficient of species i in an ideal solution is equal to the fugacity coefficient of pure species i in the same physical state as the solution and at the same T and P. Since Raoult's law is based on the assumption of ideal- solution behavior for the liquid phase, the same systems that obey Raoult's law form ideal solutions.

Excess property Definition: (6.17) For example: It follows that: (6.18)

The definition of ME is analogous to the definition of a residual property. Indeed, excess properties have a simple relation to residual properties, found by subtracting Eq. (3.41) from Eq. (6.17):

Since an ideal-gas mixture is an ideal solution of ideal gases, Eqs. (6.9) through (6.12) become expressions for Mig when Mi is replaced by Miig. For example, from eq. (6.9): (6.9) General relation: (6.19)

This leads immediately to the result: (6.20) Note that excess properties have no meaning for pure species, whereas residual properties exist for pure species as well as for mixtures. The partial-property relation analogous to Eq. (6.17) is: (6.21) where is a partial excess property

The fundamental excess-property relation is derived in exactly the same way as the fundamental residual- property relation and leads to analogous results. (6.22)

(5.1) (5.1) (6.13) (6.13) (6.15) (6.13) (6.23)

(5.1) (6.23) (6.24) (6.25)

(6.23) (6.22) (6.26) (6.27) (6.28) (6.29)

(6.27 – 6.29) (5.15 – 5.17)

(6.29) (6.30)

(6.31) (6.32) (6.32)

(6.26) (6.33)

(6.24) (6.24) (4.63) (4.63)

(6.34) (5.9) (4.63) (6.24) (6.34) (6.35) (6.36)

(6.32) (6.32) (6.37)

(6.38) (6.29) (6.29)

(6.39)

(6.39) (6.40) (6.41)

(6.40) (6.29)

(6.42) (6.43)

(6.44) (6.45) (6.46)

(6.47) (6.48) (6.49)

(6.50) (6.51) (6.52)

(6.53) (6.54) (6.55)

(6.56)

(6.57) (6.58) (6.59) (6.60)

(6.61) (6.62) (6.63) (6.64)