1.) The workdone to assemble a collection of charges is given bythe formula Wensemble = (1/2)∑i(qiVi), where Vi is the scalarpotential at the point occupied by the ith charge in the absence ofthat charge. The scalar potential is equal to the work required tomove a test charge of+1 from infinity to the location of qi in thepresence of all other charges in the ensemble. The constant ofproportionality for Wensemble is arbitrarily taken as unity. Bydoing so, the value of Wensemble does not conform to standard unitsof energy, but we are interested only as to the value (positive ornegative) of Wensemble relative to zero.
a.) The work done in moving a pair of charges from infiniteseparation to a finite distance r can be determined by evaluating apath integral; however, the formula for Wensemble should work fortwo charges as well as a dozen charges. Show that the formula forWensemble gives the same result as the path integral, namelyW=q1q2/r.
b.) Is there an equal-distant arrangement of three charges thatis of lower energy that an infinite separation of those charges? Inother words, can we assemble three charges at the vertices of anequilateral triangle that requires positive work to move thosecharges to an infinite separation? Compute Wensemble for twopossibilities.
c.) The sodium ion channel is a tetramer with four-foldrotational symmetry. Each subunit provides a line of four carbonyloxygen atoms perpendicular to the surface of the bilayer membrane.The pore selectively passes the sodium ion without the expenditureof energy, and can be represented schematically as a stack of foursquares, separated by a distance of 3.8 AÌŠ, with carbonyl oxygenatoms occupying the vertices of each square. Sodium ions putativelybind at the center ofeach square, coordinating to the carbonyloxygen atoms, but is this a stable arrangement on the basis ofelectrostatic interactions? For the second and third layers, thenearest neighbors to the sodium ions are given by the image belowto the right. The distance between oxygen atoms and the centralsodium ion is 2.5 AÌŠ. Assuming a charge of +1 for the sodium ion,what charge (sign and magnitude) must the oxygen atoms have toassemble the system on the right side with no expenditure of work?(No expenditure of work implies the sodium ion encounters nobarrier in its passage through the pore). Is this a reasonablepartial charge for the oxygen atom of a carbonyl group? Repeat thecalculation using a distance of 3.8 AÌŠ between oxygen and sodiumatoms. Can a potassium ion make it through a pore of thisdesign?
