II. Dimensions Involving Derivatives When we do dimensionalanalysis, we do something analogous to stoichiometry, but withmultiplying instead of adding.

Part A) Consider the diffusion constant that appears in Fick’sfirst law: J = −D dn dx. In this expression, J represents a flow ofparticles (the number of particles per unit area per second), nrepresents a concentration of particles (the number of particlesper unit volume), and x represents a distance. We can assume thatthey have the following dimensionalities: [J] = 1 L 2T , [n] = 1 L3 , [x] = L.

1. What is the dimensionality of dx? Explain. (Hint: Rememberfrom calculus that dx is just an infinitessimally small portion ofx.)

2. Based on part (A), what is the general rule for thedimensionality of any differential? That is, given any physicalquantity, Z, how does the dimensionality of dZ compare to that ofZ?

3. Given any two physical quantities, Y and Z, what is thedimensionality of the derivative dY dZ in terms of [Y ] and[Z]?

Part B) Einstein discovered a relation that expresses how Ddepends on the parameters of the system: the size of the particlediffusing (R), the viscosity of the fluid it is diffusing in (η),and the thermal energy parameter (kBT). Assume that we can expressD as a product of these three quantities to some power, like this:D = (kBT) a (η) b (R) c .

1. Rewrite the equation above in terms of M, L, and T, bysubstituting in your answer for the dimensionality of D fromquestion 4 above, and the dimensionalities of the other constants,which are: [kBT] = ML2 T

2 , [η] = M LT , [R] = L 2. By “balancing” M, L, and T on eachside of the expression above, determine three equations in terms ofa, b, and c that will guarantee that D will have the correctdimensionality.

3. Solve these equations and write an expression for how Ddepends on the three parameters. (The correct equation has a factorof 1/6Ï€ that cannot be found from dimensional analysis.)