Vhen we do dimensional analysis, we do something analogous to stoichiometry, but with multiplying instead of adding. Consider the diffusion constant that appears in Fick's first law: dn J=-D dx n this expression, J represents a flow of particles: number of particles per unit area per second, n represents a concentration of particles: number of particles per unit volume; and x represents a listance. We can assume that they have the following dimensionalities: • ) = 1/L2T • [n] = 1/L3 • [x] = L From this, determine the dimensionality of D. (h) Einstein discovered a relation that expresses how D depends on the parameters of the system: the size of the particle diffusing (R), the viscosity of the fluid it is diffusing in w,and the thermal energy parameter (kg 7). These have the dimensionalities • [kg 7] = ML2/T2 • ) = M/LT • [R] = L Assume that we can express Das a product of these three quantities to some power, like this: D= (kg na (b (R)C Vrite eauations for a, b, and c that will guarantee that D will have the correct dimensionality for M, L, and T. (i) In this case, there were exactly the right number of parameters to give the same number of equations as there were unknowns. That will not always be the case, but sometimes his method still can be made to work by adding physical knowledge about the dependences. Solve these equations and write an expression how D depends on the three parameters. (The correct equation has a factor of 1/(6n) that cannot be found from dimensional analysis. Make sure to nclude this factor in vour answer.)
Vhen we do dimensional analysis, we do something analogous to stoichiometry, but with multiplying instead of adding. Consider the diffusion constant that appears in Fick's first law: dn J=-D dx n this expression, J represents a flow of particles: number of particles per unit area per second, n represents a concentration of particles: number of particles per unit volume; and x represents a listance. We can assume that they have the following dimensionalities: • ) = 1/L2T • [n] = 1/L3 • [x] = L From this, determine the dimensionality of D. (h) Einstein discovered a relation that expresses how D depends on the parameters of the system: the size of the particle diffusing (R), the viscosity of the fluid it is diffusing in w,and the thermal energy parameter (kg 7). These have the dimensionalities • [kg 7] = ML2/T2 • ) = M/LT • [R] = L Assume that we can express Das a product of these three quantities to some power, like this: D= (kg na (b (R)C Vrite eauations for a, b, and c that will guarantee that D will have the correct dimensionality for M, L, and T. (i) In this case, there were exactly the right number of parameters to give the same number of equations as there were unknowns. That will not always be the case, but sometimes his method still can be made to work by adding physical knowledge about the dependences. Solve these equations and write an expression how D depends on the three parameters. (The correct equation has a factor of 1/(6n) that cannot be found from dimensional analysis. Make sure to nclude this factor in vour answer.)
University Physics Volume 1
18th Edition
ISBN:9781938168277
Author:William Moebs, Samuel J. Ling, Jeff Sanny
Publisher:William Moebs, Samuel J. Ling, Jeff Sanny
Chapter1: Units And Measurement
Section: Chapter Questions
Problem 10CQ: For each of the following scenarios, refer to Figure 1.4 and Table 1.2 to determine which metric...
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