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.)
College Physics
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Chapter1: Units, Trigonometry. And Vectors
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Part g, h and i, please?
Transcribed Image Text:One of the most important reactions in biology is photosynthesis. In this complex chemical process, plants combine molecules of carbon dioxide and water
to create glucose and oxygen. The reaction looks something like this:
CO2 + H20 – C6H1206 + 02
Of course the number of atoms don't balance on the two sides. You might be able to intuit your way to the correct counting (stoichiometry), but let's do it systematically by setting
up equations.
(a) Write the reaction as
a CO2 + b H20 -→ c CGH1206 + d O2
where a, b, c, and d are meant to be integers adjusted so that the number of each kind of atoms balance on each side. For this reaction, we have three kinds of atoms that must balance: carbon (C),
hydrogen (H), and oxygen (0). Write three equations for a, b, c, and d that, if satisfied, will guarantee that the number of atoms of each kind are the same on each side of the reaction.
(b) Find the smallest set of integers that will satisfy these equations. First, solve for a.
(c) Solve for b.
![(c) Solve for b.
(d) Solve for c.
(e) Solve for d.
(f) Write the correct stoichiometric equation for photosynthesis.
(g) Dimensional analysis
When 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
In 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
distance. We can assume that they have the following dimensionalities:
• ) = 1/L?T
• [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
(1),and the thermal energy parameter (kg7). 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= (kg7)a (b (R)
Write 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
this 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
include this factor in your answer.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9be67d84-1d17-4378-951f-d2d7b08ac873%2Fd1f84ee7-a0d5-4818-9b95-a17115709a37%2Fouhkkv9i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(c) Solve for b.
(d) Solve for c.
(e) Solve for d.
(f) Write the correct stoichiometric equation for photosynthesis.
(g) Dimensional analysis
When 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
In 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
distance. We can assume that they have the following dimensionalities:
• ) = 1/L?T
• [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
(1),and the thermal energy parameter (kg7). 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= (kg7)a (b (R)
Write 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
this 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
include this factor in your answer.)
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