Absorption from a Bubble with Reaction Consider a spherical bubble of radius R(t) containing pure gas A. The gas diffuses into the surrounding liquid, where it is consumed by a first-order, irreversible, homogeneous reaction (rate constant k ). The liquid concentration of A at the gas-liquid interface is C0. Assume that CA=0 far from the bubble and that the liquid is stagnant. (a) If R(t) varies slowly enough and if enough time elapses following introduction of the bubble, a pseudosteady analysis is justified for the liquid. Derive expressions for the liquid concentration profile, CA(r,t), and the flux of A at the gas-liquid interface, NA,r(R,t). (Hint: The transformation CA(r,t)=(r,t)/r will be helpful. (b) In part (a), convective transport caused by bubble shrinkage was neglected. Identify the conditions under which this is a good approximation. Specifically, what must be the relationship between C0 and the concentration of A in the gas bubble (Cb) ? (Hint: Use an interfacial balance to relate Cb,C0dR/dt, and the flux of A in the liquid.) (c) Use order-of-magnitude reasoning to identify the range of times for which a pseudosteady analysis is justified, Assume that the conditions of part (b) are satisfied (i,e., convection is negligible)