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Homework answers / question archive / 1)A circular capillary has a diameter of 10 μm and a length of 3 cm

1)A circular capillary has a diameter of 10 μm and a length of 3 cm. The surface carries a zeta potential of −0.5 V. The flowing fluid is an aqueous solution of 0.01 M CaSO4, and the viscosity is 0.001 Pa ·s at the specified temperature. Find the volumetric flow rate. Also compute the pressure gradient that would need to be applied to create the same flow if no electric field were applied.

2)Find the flow rate for an electro-osmotic flow in a tube of diameter 5 μm and length 10 cm filled with an aqueous solution. The applied potential is 1 kV. The zeta potential is given as −100 mV. Answer: Q = 1.39 × 10−14 m3/s.Find the streaming potential when a pressure drop of 0.1 MPa is applied to a liquid with a specific conductivity of 0.0014 S/m. The surface charge is 100 mV.

3)A solid is in the form of a long cylinder and has a surface charge of qS. Derive an expression for the potential in the external region. Use the Debye–Hückel approximation and solve as a linear differential equation.A solid is in the form of a sphere and has a surface charge of qS. Derive an expression for the potential in the external region.

4)Calculate the Debye length for a 0.1-M solution of a univalent (1, 1) electrolyte, i.e., one with single-charged cations and single-charged anions, in water. If the surface has a negative charge with a zeta potential of 100 mV, plot the concentrations of the co-ions and counterions near the electrode surface. The co-ion is defined as the ion with the same charge as the electrode. Repeat for a divalent (2,2) electrolyte. Which case has the larger Debye length?

5)Consider a charged membrane of thickness 100 μm with concentrations of 1 M HCl on one side and 0.1 M HCl on the other side. The membrane has a negative charge with a constant electric field of 1 N/C. Find the flux of HCl in the system using the model for transport across a charged membrane given in Section 22.3.

6)Consider an uncharged membrane of thickness 100 μm with concentrations of 1 M HCl on one side and 0.1 M HCl on the other side. Assume that HCl is completely ionized. (a) Find the flux of HCl in the system. (b) Find the diffusion potential generated in the system. (c) Which side of the membrane is at a higher potential? Explain your answer in terms of the physics of the system. (d) Find the diffusion flux and migration flux of Cl− and H+ ions and tabulate them. Comment on the results.

7)Calculate the diffusion potential for an uncharged membrane with a concentration of 0.5 M on one side and 0.1 M on the other side for solutions of (a) CuSO4, (b) MgCl2, and (c) KCl. Also calculate the effective diffusion coefficients for these systems.

8)Estimate the steady-state concentration profile when a typical albumin solution is subjected to a centrifugal field of 30 000 times the force of gravity under the following conditions: the cell length is 1.0 cm, the molecular weight of albumin is 45 000, the apparent density of albumin in solution is 1.34 g/cm3, and the mole fraction of albumin at z = 0 (one end of the cell) is 5 × 10−6.

9)Derive the solutions for transient concentration profiles in the two-bulb apparatus (Example 21.5 in the text) for the binary case, and show that the multicomponent case can be derived as an extension of this. What assumption is implicit in extending the binary case to the multicomponent case?

10)Composition dependence of the Fick matrix K˜ . Consider the three-component system consisting of acetaldehyde (1), hydrogen (2), and ethanol (3). The binary diffusivity values at 548 K and 101.3 kPa are given in Example 21.3. Find the K matrix at the bulk gas composition and at the catalyst surface concentration. Use ethanol as the component eliminated (the solvent). Comment on the values and the extent of multicomponent interactions.