Chemical Potential

8. Chemical Potential

The chemical potential drives mass (or species) transfer in

a manner similar to the thermal potential that drives heat transfer from higher to lower temperatures.

a. Multicomponent into Mul- ticomponent

Consider a vessel divided Figure 26. Heat transfer mechanism. into two sections C and D (as shown in Figure 27 ) that initially contains oxygen throughout, and in which charcoal is spread over the floor of section D. As- sume that as the charcoal is burned, sections C and D consist of two components: oxygen and

CO 2 . Further, consider a specific time at which the mole fraction of O 2 in section C (say, x O 2 = 80%) is larger compared to that in section D (say, X O2 = 30%).

Since molecules move randomly, for every 1000 molecules that migrate from C into

D through the section Y–Y, 1000 molecules will move from D into C. Consequently, 800 molecules of O 2 will move into D while only 300 molecules of this species will migrate to C from D, so that there is net transfer of 500 molecules of O 2 from section C into D. Simultane- ously, there is a net transfer of 500 molecules of CO 2 across the Y–Y plane from section D into

C. The oxygen transfer enables continued combustion of the charcoal. This mass transfer (or species transfer) due to random molecular motion is called diffusion. The chemical potential µ for ideal gases is related to the species concentrations

(hence, their mole fractions). A higher species mole fraction implies a higher chemical poten- tial for that species. For instance, the chemical potential of O 2 , µ O 2 is higher in section C com-

pared to D, thereby inducing oxygen transfer from C to D. If the charcoal is extinguished, CO 2 production (therefore, O 2 consumption) ceases, and eventually a state of species equilibrium is reached. At this state the chemical potential of each species or its concentration is uniform in the system.

b. Single Component into Multicomponent Consider the following scenario. A vessel is divided into two sections E and F by a porous membrane, as shown in Figure 28a . Section E initially contains a single component (denoted by o) at a lower pressure, and Section F contains a multicomponent gas mixture at the same temperature, but at double the pressure. Assume that the mole fraction of o molecules in section F is initially x o,F = 0.2, and that there are 50 molecules per unit volume contained in section E and 100 molecules per unit volume in section F. Further, assume the porosity of the membrane to be selective such that it allows only o molecules to be transferred through its pores (i.e., it is a semipermeable membrane). Assuming 200 molecules s –1 of o to migrate from

E into F, 400 molecules of all species will attempt to transfer into E from F due to the higher pressure in that section. However, the semipermeable membrane allows only o molecules to

CO 2 D

Figure 27: Illustration of species transfer. Oxygen mole-

cules are denoted by o and CO 2 molecules by x.

transfer from F, so that of these 400 only the 80 molecules of o move from F into E. Therefore, there is net flow equal to (200–80)=120 molecules s –1 from E into F. If the pressure in section

F is increased eightfold, molecules of species o can no longer be transferred into it, since of the 1600 molecules that now attempt to migrate every second, the membrane allows only the 320 which are of o to move into section E (cf. Figure 28b ). The net motion is 320 – 200 = 120 molecules s –1 into section E from F.

Therefore, by adjusting the pressure in section F, we can control the direction of spe- cies transfer, or prevent it altogether by maintaining chemical equilibrium. For example, if under these conditions, the pressure in section F is five times that in E, 1000 molecules s –1 at- tempt to migrate from F to E, but only 200 molecules s –1 of o actually do, balancing the trans- fer of the same amount from E to F. The chemical potential of species o becomes uniform across the membrane at this state. Altering the pressure from this condition will change the chemical potential. In general, the larger the pressure, the higher the chemical potential.