ChE 541 Homework 4 Fall 2012Mass Transfer Due 10/9/12 M. Sahimi1. Transport Through a Charged Hydrogel MembraneSome membranes are made of hydrogels, which are a particular typ e of material in whichover 90% of the volume is water. The membrane is held together by crosslinking ofthe polymer’s monomers. Consider such a charged membrane of thickness L in whichthe concentration of the fixed (immobile) negative charges is Cm. The membrane isconnected on both sides to salt solutions where at z = 0 C+= C−= C0, while at z = L,C+= C−= CL. The diffusivities D+and D−are given, but the charge potential φ withinthe membrane is not known. Assume the system to be one dimensional. Recall that thenet electrical current is zero.(a) The implication of the assumptions for the fluxes is that, N+= N−= Ns, where Nsis the common value of N+and N−. Why?(b) Write down the expression for Ns. Due to (a), one can write the expression in termsof C−and D−, or C+and D+. Give both expressions.(c) The electroneutrality condition implies that, within the membrane, C+(z) = C−(z) +Cm. Why?(d) Use (c) to eliminate C+from the equation in (b) to obtain an equation for C−. Then,use this equation to eliminate the potential φ from the equation in (b).(e) Solve the final equation for C−obtained in (d), and give the expression for Nsthatdoes not contain φ. The expression should contain C−(0) and C−(L).(f) To be practical, we need to relate C−(0) and C−(L) to C0and CL, which are known(as well as measurable). To do so, proceed as follows. First write down the generalexpression for the flux Niof a comp onent i, in terms of the diffusive, convective, andmigration contributions (given in the class). Then, use the fact that vi= 0, and Ji= 0,to show that,ln Ci+ziFRTφ = constant . (A)Use this to show that C+C−is a constant. Thus, show that, C+(0)C−(0) = C20, andC+(L)C−(L) = C2L.(g) Now, solve for C−(0) and C−(L) in terms of Cm, CL, and C0.(h) Using (g), derive an expression for Ns.(i) Use Eq. (A) to derive an equation for φ(0), assuming that the external potential isφ0. A similar expression can be derived for φ(L).2. Diffusion of Water and Ammonia Through a MembraneA mixture of ammonia (component A) and water (component B) vapors passes through aslab of thickness h that contains helium (compoent C). The slab represents a membrane,and the passage of the two types of vapor is intended for their separation. A and B enterthe slab through one face and leave from the opposite face with mole fractions xA1andxB1, and xA2and xB2, respectively.(a) To be practical one must relate the fluxes of NAand NBto the boundary conditions,where the mixture’s composition is known and/or can b e measured. Assume that thethree diffusivities DAm, DBmand DCmare known, where m represents the mixture, andthat the transp ort processes is at steady state. Derive expressions for NAand NB, ifhelium is stationary.(b) Rework the problem if, instead of the diffusivities of the gases in the mixture, we onlyknow the binary diffusivities DAB, DBCand DAC. Is there a difference with the resultsin (a)? Why?3. Fast Reaction of Radicals in Liquid SolutionsWhen free radicals, such as NO and O−2, are in a liquid, they react so fast that canbe considered essentially as instantaneous. Determining the reaction rate constant forsuch cases is not easy, and is usually based on diffusion measurement. We assume thatmolecules A and B that are spheres of radii a and b react immediately as soon as theycome into contact; that is to say, when the distance between their centers is a+b. Assumesteady state and use spherical coordinates.(a) We assume that A is stationary, and is surrounded by diffusing molecules of B. IfCB= CB∞far from A, determine the concentration profile CB(r) and the rate of reactionper molecule of A.(b) On e can show that if A is also mobile, the effective diffusivity is DA+ DB. Determinethe rate of reaction for this case.(c) Determine the volumetric rate of reaction RAfor (b). If we define an effective reactionrate constant k by, RA= RB= −kCA∞CB∞, derive an expression for k, where CA∞isthe bulk concentration of