Question

In: Physics

II. Dimensions Involving Derivatives When we do dimensional analysis, we do something analogous to stoichiometry, but...

II. Dimensions Involving Derivatives When we do dimensional analysis, we do something analogous to stoichiometry, but with multiplying instead of adding.

Part A) Consider the diffusion constant that appears in Fick’s first law: J = −D dn dx. In this expression, J represents a flow of particles (the number of particles per unit area per second), n represents a concentration of particles (the number of particles per unit volume), and x represents a distance. We can assume that they have the following dimensionalities: [J] = 1 L 2T , [n] = 1 L 3 , [x] = L.

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

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

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

Part B) 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 (η), and the thermal energy parameter (kBT). Assume that we can express D 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, by substituting in your answer for the dimensionality of D from question 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 each side of the expression above, determine three equations in terms of a, b, and c that will guarantee that D will have the correct dimensionality.

3. Solve these equations and write an expression for how D depends on the three parameters. (The correct equation has a factor of 1/6π that cannot be found from dimensional analysis.)

Solutions

Expert Solution

Let us assume Diffusion coefficient D = ( kB T)ab Rc .....................(1)

where is dimensionless constant, kBT is thermal energy, kB is Boltzman constant, T is temperature,

is coefficient of viscosity and R is size of diffusing particle .

Diffusion coefficient D is given as

J = -D ( dn/dx ) ..............(2)

where J is the rate of particle flux , i.e., number of particles per unit area per unit time , (dn/dx) is the spatial gradient of density, i.e number of particles per unit volume per unit distance.

Let us get the dimension of D from eqn.(2)

[D] = [J] [dx] / [ dn ] = [ L-2 T-1 ] [ L ] / [ L-3 ] = [ L2 T-1 ] ...................(3)

Using dimensional formula of diffusion coefficient D as given in eqn.(3), and applying respective dimensional formula for thermal energy, coefficient of viscosity and size in eqn.(1), we get the dependence power a, b, c as given below

dimension of thermal energy kBT = [ ML2 T-2 ]

dimension of viscosity = [ M L-1 T-1 ]

dimension of size R = [ L ]

Dimensional analysis of eqn.(1) :-   [ L2 T-1 ] = [ ML2 T-2 ]a [ M L-1 T-1 ]b [L]c

equating power of [ M ] on both side, we get , a + b = 0    ....................(4)

equating power of [ L ] on both side, we get, 2a - b + c = 2 ...................(5)

equating power of [ T ] on both side, we get, 2a + b = 1 ...................(6)

By solving equations (4), (5) and (6), we get a =1 , b = -1 and c=-1

Hence from eqn.(1), we get D = ( kBT ) / ( R )

If we assume the constant = (1/6) , then D is given as


Related Solutions

In general, how do we calculate the value of something?
In general, how do we calculate the value of something?
Now that we are discussing OLAP and dimensions, the question is: Why do we want our...
Now that we are discussing OLAP and dimensions, the question is: Why do we want our data broken down? What are the benefits?
Why is the analysis of historical accounting information relevant when we do valuations?
Why is the analysis of historical accounting information relevant when we do valuations?
What are the five dimensions we rely on when evaluating the quality of a service? Responsiveness,...
What are the five dimensions we rely on when evaluating the quality of a service? Responsiveness, Insurance, Telepathy, Intangibles, Responsibility Reliability, Responsiveness, Assurance, Empathy, and Tangibles Reliable, Intangible, Persuasive, Attractive, Perishable Touchability, Movability, Reliability, Identifiability, Exceptionality What is the most important aspect of planning and conducting a useful experiment? Effective Stimuli Causality Good Questions Random Assignment
In the Bakalar case involving artwork stolen during World War II, do you agree with the
In the Bakalar case involving artwork stolen during World War II, do you agree with the court’s decision? Should the heirs get a chance to recover the drawing that was stolen from their ancestor? Or should Bakalar, who has owned the drawing for fifty years and knew nothing about its origin, be able to keep ownership?  
Critically do an Impact analysis of COVID-19 (Corona Virus) on Derivatives Investments (DI) and formulate a...
Critically do an Impact analysis of COVID-19 (Corona Virus) on Derivatives Investments (DI) and formulate a Business plan post COVID-19 (next 7 months) with the aim of achieving 6 main Investment objectives for DI?
Is critical thinking something that we would use all the time or only when the situation...
Is critical thinking something that we would use all the time or only when the situation warrants it? How do you go about making the decision to use critical thinking? In other words, what are the conditions for its use?
Do a character analysis of Nora Helmer in Ibsen’s A Doll House. Say something clear and...
Do a character analysis of Nora Helmer in Ibsen’s A Doll House. Say something clear and concise about Nora’s personality and then defend it.
A-Why do we use electronic and computerized instrumentation in this laboratory, and not something less complex?...
A-Why do we use electronic and computerized instrumentation in this laboratory, and not something less complex? B-explain the main difficulty that arises with the use of electronic and computerized instrumentation. Why do you think it is important to acknowledge this difficulty? C-Again, in your own words, explain the two things you need to do in order to get good data when using computerized (or any other) instrumentation. D-The Motion Sensor works on the principle of echolocation. Describe how the sensor...
How do we overcome the stereotype that mentally ill patients are ill because of something they...
How do we overcome the stereotype that mentally ill patients are ill because of something they did while physically ill patients are ill because of something that happened to them? Even physical illness is often blamed on the patient (e.g., lifestyle behaviors), (1) to what extent does such blaming interfere with, even suppress, empathy or patient-centered care? (2) How do insurance restrictions reduce the quality of care and diminish outcomes for mentally ill or lifestyle-related illness?”
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT