代写2024 - PHY 4390-8191 (Separation Science)代做Statistics统计
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1. BONUS QUESTION – True or False
(+1 for the right answer; -1 for a wrong answer; 0 if no answer)
Minimum 0 bonus points (cannot be negative). Be strategic!
(a) Polymeric gels are crosslinked using covalent bonds.
(b) The Bjerrum length limits the density of charges on glass walls.
(c) Polymer coatings (brushes) made of charged polymers can generate EOF.
(d) Taylor dispersion can be used to mix liquids in microluidic devices.
2. Deinitions
Briely describe the following items and explain their use in Separation Science
Total = 4 points
(a) The Kurtosis.
(b) Entropic Trapping.
(c) The luctuation-dissipation theorem.
(d) The P,eclet number.
3. An big protein in water
Total = 4 points
Its difusion coeicient is D = 6.90 10-7 while its mass is 64 000 Daltons. Assume that it is a perfect sphere for this problem.
(a) Estimate its radius using Stokes’s law.
(b) The mass density of the protein is ρ = 1.33 (atypical value for proteins). On that basis, what should be its radius?
(c) If you ind a discrepancy between your two previous results, perhaps it is because the protein is surrounded by a ”hydration layer”, a hydration shell of water that is attached to the protein and follows it when it difuses.
Estimate the thickness of this shell (both in nm and number of layers of water molecules) given your previous two results.
4. We study a 250 kbp (kilo-basepairs) dsDNA molecule in solution
Total = 5 points
At every step, make sure you justify the choice of equations you make.
(a) Estimate its radius-of-gyration Rg .
(b) Is it much bigger than the Kuhn length and contour length of the molecule?
(c) Inside this sphere of radius Rg , what fraction of the volume is occupied by the solvent?
(d) Estimate its difusion coeicient D.
(e) Estimate its relaxation time.
5. Radius-of-gyration vs hydrodynamic radius. Polymer vs hard spheres.
Total = 7 points
We have an ideal random-walk polymer with N monomers of size b.
It is fully lexible: bK = b.
(a) What is its radius-of-gyration Rg (N)?
(b) What is its hydrodynamic radius RH (N)?
(c) Which one is the smallest? Why?
(d) We now have a hard sphere with a radius r = RH (N).
Which one will have the lowest difusion coeicient: the polymer or the hard sphere?
(e) What is the radius-of-gyration rg of the hard sphere?
(f) What is the hydrodynamic radius rH of the hard sphere?
(e) In the case of the sphere, which one is the largest, rH or rg?
6. Comparing electrophoretic mobilities:
Total = 5 points
We carry out free solution electrophoresis in a capillary of length L.
The walls of the capillary are charged. The corresponding surface potential is (0) = 250 mV.
(a) If you use a salt containing only monovalent ions (such as KCl), at what concentration is the Debye length equal to the Kuhn length of dsDNA?
(b) Instead, we want to use a Debye length of λD = 12 nm.
What salt concentration do we need?
(c) Estimate the electrophoretic mobility of the following ions: K+ , Cl- .
(d) Estimate the EOF mobility in the capillary.
(e) Considering the previous result, do you think you may face problems if you want to elec- trophorese DNA in this capillary?
7. The Capture radius of large DNA molecules
Total = 5 points
We have a nanopore translocation device with a pore of radius rp = 2 nm and length `p = 5 nm.
The applied voltage is ΔV = 250 mV.
(a) Estimate the ratio between the capture radius R* and the hydrodynamic radius RH of a large DNA molecule with M 参 1 basepairs.
(b) Do you think that we can treat the DNA molecule as a point-like object to understand the physics of the capture process?
8. Steric FFF
Total = 10 points
In Section 6.2.1, we studied steric FFF between two parallel walls.
(a) Redo these calculations for particles of radius r inside a cylindrical tube of radius R > r.
(b) I deine the fractional net velocity of a particle of size r as ν(r) = .
Plot ν(r) .
(c) The derivative κ(r) = · informs us about performance of an FFF elution system of length L.
i. What interpretation do you give to κ(r)?
ii. Which system is the most promising: the 2-wall system or the cylindrical one? iii. Are these systems better for small (r 冬 R) or large particles?
9. Volume of a polymer in a small tube
Total = 10 points
A lexible polymer with N 》 1 monomers of size b is inside a nanotube of diameter d.
(a) How does the polymer’s radius-of-gyration Rgo varies with N and b when it is NOT inside the nanotube? What is the volume Vo occupied by the polymer in this case?
(b) We have seen how the chain extension Rjj (N, d, b) can be estimated using scaling argu- ments, for example in Section 12.4.2.
(c) Calculate the volume Vn (N, d, b) occupied by the polymer when it is inside the tube. Is Vn larger or smaller than Vo?
(d) The chain extension Rjj (N, d, b) is a good estimate of the radius-of-gyration along the
tube axis. In the transverse direction, the radius of gyration is trivially R? d.
Plot the global radius-of-gyration Rg(2) = Rjj(2) + 2R?(2) vs d for N = 1000 and b = 1.
(e) Is the radius-of-gyration calculated in (c) above a monotonic function of d? If not, for what value of d do you ind an optimum? What is the value of Rg under such conditions?
Figure 20.1:
————
(a) The solid surface is charged (+) here.
(b) The Electric potential 亚(r), or φ(z) in this drawing, decays exponentially, with a characteristic length scale λD , the Debye length. (c) The concentration of ions is either increased (counterions – or anions here) or depleted (coions or cations here) near the wall.
10. A charged wall
Total = 10 points
The wall surface charge density is σ (charge per surface area) and the wall potential is Ψ(0).
The bufer contains a monovalent salt concentration ρo.
Calculate the relation between Ψ(0) and σ in the low potential limit.
Hint: The net charge of the liquid solution must neutralize the wall charge!