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Unformatted text preview: Physics 112 Fall 2010 Professor William Holzapfel Homework 6 Solutions Problem 1, Kittel 5.1: Centrifuge When the gas is in equilibrium, we know that the total chemical potential will be constant through- out. The total chemical potential is = int + ext , and we know that the internal chemical potential (of an ideal gas) is related to the number density by = ln n ( r ) n Q , so we see that a spatial variation in the external chemical potential will create a spatially varying number density. A rotating gas feels a centrifugal force F = Mv 2 /r = Mr 2 . We can consider this force to be caused by an effective potential such that F =- dV/dr : ext ( r ) = V ( r ) =- Z drF ( r ) =- M 2 r 2 2 . The total chemical potential ( r ) = ln n ( r ) n Q- M 2 r 2 2 will be constant throughout the centrifuge in equilibrium, so we can equate the chemical potential at a radius r with the chemical potential at the axis of rotation: (0) = ln n (0) n Q = ln n ( r ) n Q- M 2 r 2 2 = ( r ) Solving for n ( r ) we have: n ( r ) = n (0) exp Mr 2 2 2 (1) So we see that we get an exponentially higher concentration of particles at larger r (as expected). Problem 2, Kittel 5.4: Active Transport We know that the ratio of the concentration of K + is n sap /n water = 10 4 , so we can find the difference in the chemical potentials of K + of the sap and the water by treating the ions as an ideal gas: 4 = sap- water = ln n sap n Q- ln n water n Q = ln n sap n water = ln(10 4 ) = (300 K ) k b * 4 ln(10) = (1200 K) ( 1 . 38...
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