# What is the surface density

BIOE2397 Homework #6

(Homework #6 is due on Wednesday, 12/5, 11:59pm)

Problem 1 [5 points]: If resting membrane potential of a cardiac cell is -90mV, the concentration of πΎπΎ+ outside the cell is 5mM and membrane permeability for all cations (except for πΎπΎ+) are zero. What is the concentration of πΎπΎ+ inside of the cell?

Problem 2 [10 points]: A cellβs resting membrane potential is -50mV, and the cellβs diameter is about 10ΞΌm. The capacitance of the cell membrane is about 0.5 ΞΌF/cm2, and the internal concentration of πΎπΎ+ is 150mM. What is the maximum surface density of πΎπΎ+ in the cell? What is the surface density of πΎπΎ+ needed to produce the resting membrane potential?

Problem 3 [25 points]: The relative importance of advection versus diffusion is described by a non-dimensional parameter called the Peclet number, ππππ = ππ(π’π’,π·π·, π₯π₯), where π’π’ is the velocity, π·π· is the diffusion coefficient, and π₯π₯ is the distance to the point of interest. Using dimensional analysis, find the form of ππππ such that ππππ βͺ 1 is advection dominated, and ππππ β« 1 is diffusion dominated. For stream with π’π’ = 0.3 m/s and π·π· = 0.05 m2/s, where are diffusion and advection equally important?

Problem 4 [40 points]:

A student injects 5 mL of 20% Rhodamine solution in water (with specific gravity 1.15 g/mL) instantaneously and uniformly over the tube cross-section (π΄π΄ = 0.8 cm3) at the point π₯π₯ = 0 and the time π‘π‘ = 0. The tube is filled with stagnant water. Assume the molecular diffusion coefficient if π·π· = 0.13Β·10-4 cm2/s.

β’ What is the concentration at π₯π₯ = 0 at the time π‘π‘ = 0?

β’ What is the standard deviation of the concentration distribution π‘π‘ = 1 second after injection?

β’ Plot the maximum concentration in the tube, πΆπΆππππππ(π‘π‘), as a function of time over the interval π‘π‘ = [0, 24 h].

β’ How long does it take until the concentration over the region π₯π₯ = [-1, +1] meters can be treated as uniform? (A uniform distribution is one where the minimum concentration within a region is no less than 95% of the maximum concentration within that same region.)

BIOE2397 Homework #6

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