Question

The capacitance of an object is a measure of how much charge separation can be produced on the object by applying a voltage difference to it, or contrarily, how much voltage arises across the object as a result of a particular charge separation developing. The equation defining the capacitance is

Q = C ΔV

where Q is the charge separation (+Q and -Q at two different places on the capacitor), C is the capacitance, and ΔV is the potential difference. But you can also read this equation from right to left: The capacitance also identifies how much voltage difference arises across the object as a result of a particular charge separation Q.

PART A: The unit we use for the capacitance is the Farad = 1 Coulomb/Volt. What is the dimensionality of the unit Farad (in terms of M, L, T, and Q -- mass, length, time, and charge)? (If you are not familiar with this idea, check out the pre-requisite webpages.) Express the unit Farad in terms of basic SI units (kilogram, meter, second, Coulomb).

A simulation of a lipid membrane imbedded in water (blue). Pastor, Venable, & Feller, Acc. Chem. Res. (2002) 35, 438-446.

PART B: A typical cell membrane in an animal maintains a potential difference across the membrane of ΔV = 70 mV and the membrane has a thickness of about 8 nm. The capacitance of the membrane is about 1 microFarad per square cm.  If we model the membrane by a simple "two thin sheets of charge" model separated by 8 nm with nothing between them, what would be the electric field be between the sheets and what would the charge density on the sheets of the membrane? Explain your reasoning.

PART C: Clearly, a membrane is not "empty" -- it's filled with lipid molecules as shown in the figure at the right. On the average, the dielectric constant, kappa, of the membrane is about 3. How would that change the average E field and the surface charge density that you found in part (B)?

PART D: In order to maintain the potential difference across the two sides of the membrane, cells carry ions across a membrane against the field ("uphill") using a variety of active transport mechanisms. One mechanism does so by using up some of the cell's stored energy converting ATP to ADP. How much work does it take to carry one sodium ion (charge = +e) across the membrane against the field? Calculate your answer in eV/ion, Joules/ion, and kilcalories/mole (the last for 1 mole of sodium ions).

Answer 1