Question

1.) The workdone to assemble a collection of charges is given by the formula Wensemble = (1/2)∑i(qiVi), where Vi is the scalar potential at the point occupied by the ith charge in the absence of that charge. The scalar potential is equal to the work required to move a test charge of+1 from infinity to the location of qi in the presence of all other charges in the ensemble. The constant of proportionality for Wensemble is arbitrarily taken as unity. By doing so, the value of Wensemble does not conform to standard units of energy, but we are interested only as to the value (positive or negative) of Wensemble relative to zero.

a.) The work done in moving a pair of charges from infinite separation to a finite distance r can be determined by evaluating a path integral; however, the formula for Wensemble should work for two charges as well as a dozen charges. Show that the formula for Wensemble gives the same result as the path integral, namely W=q1q2/r.

b.) Is there an equal-distant arrangement of three charges that is of lower energy that an infinite separation of those charges? In other words, can we assemble three charges at the vertices of an equilateral triangle that requires positive work to move those charges to an infinite separation? Compute Wensemble for two possibilities.

c.) The sodium ion channel is a tetramer with four-fold rotational symmetry. Each subunit provides a line of four carbonyl oxygen atoms perpendicular to the surface of the bilayer membrane. The pore selectively passes the sodium ion without the expenditure of energy, and can be represented schematically as a stack of four squares, separated by a distance of 3.8 Å, with carbonyl oxygen atoms occupying the vertices of each square. Sodium ions putatively bind at the center ofeach square, coordinating to the carbonyl oxygen atoms, but is this a stable arrangement on the basis of electrostatic interactions? For the second and third layers, the nearest neighbors to the sodium ions are given by the image below to the right. The distance between oxygen atoms and the central sodium ion is 2.5 Å. Assuming a charge of +1 for the sodium ion, what charge (sign and magnitude) must the oxygen atoms have to assemble the system on the right side with no expenditure of work? (No expenditure of work implies the sodium ion encounters no barrier in its passage through the pore). Is this a reasonable partial charge for the oxygen atom of a carbonyl group? Repeat the calculation using a distance of 3.8 Å between oxygen and sodium atoms. Can a potassium ion make it through a pore of this design?

Answer 1

1. Here below are the solution for I & II part of question only.