Consider an aqueous solution of N = 1022 dye molecules separated from pure solvent by a movable, semipermeable plug: Lmax In class we estimated the probability distribution P(L) x (L) for finding the plug at a distance L from the top of the solution, where 2(L) x LN is the number of ways available space above the plug. arrange N solutes in the If no external force is applied to the plug, we argued, the maximum value of P(L) should occur when L takes on its largest possible value, Lmax. Here you will estimate the size of typical fluctuations in plug position, AL Lmax - L. (a) What displacement ALtyp gives a probability (per unit length) equal to ten percent of the max- imum? In other words, solve the equation P(Lmax - ALıyp) = 0.1P(Lmax). For this purpose use a Taylor expansion of In 2 around Lmax To determine a numerical value for ALtyp, take the container's height to be Lmax = 10 cm. (b) Thermodynamics concerns the behavior of average macroscopic quantities, such as the length L in our dye solution, and neglects fluctuations about mean values. Comment on the validity of this approach in light of your answer to part (a).

Consider an aqueous solution of N = 1022 dye molecules separated from pure solvent by a movable, semipermeable plug: Lmax In class we estimated the probability distribution P(L) x (L) for finding the plug at a distance L from the top of the solution, where 2(L) x LN is the number of ways available space above the plug. arrange N solutes in the If no external force is applied to the plug, we argued, the maximum value of P(L) should occur when L takes on its largest possible value, Lmax. Here you will estimate the size of typical fluctuations in plug position, AL Lmax - L. (a) What displacement ALtyp gives a probability (per unit length) equal to ten percent of the max- imum? In other words, solve the equation P(Lmax - ALıyp) = 0.1P(Lmax). For this purpose use a Taylor expansion of In 2 around Lmax To determine a numerical value for ALtyp, take the container's height to be Lmax = 10 cm. (b) Thermodynamics concerns the behavior of average macroscopic quantities, such as the length L in our dye solution, and neglects fluctuations about mean values. Comment on the validity of this approach in light of your answer to part (a).

Modern Physics

3rd Edition

ISBN:9781111794378

Author:Raymond A. Serway, Clement J. Moses, Curt A. Moyer

Publisher:Raymond A. Serway, Clement J. Moses, Curt A. Moyer

Chapter11: Molecular Structure

Section: Chapter Questions

Problem 12P

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