che359_sp10_exam1_soln - Page 1 of 4 ChE 359 Exam#1...

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Unformatted text preview: Page 1 of 4 ChE 359 Exam #1 Solutions prepared by Professor Lo 1. (45 points) The function of cells and tissues is often governed by the mechanical stresses on proteins, polysaccharides, and DNA. To study these functions, engineers often use atomic force microscopy, which is a single-molecule measurement technique, to stretch these biomolecules under physiological (i.e., aqueous) conditions. In this problem, a single protein molecule (titin) i is being stretched at constant temperature between the tip of a microscopic cantilever and a flat, gold- covered substrate. The pressure of the environment is kept constant. The forces acting on the molecule as it is extended are transmitted to the cantilever, and a force vs. extension curve is generated. a. (10 points) Identify the system in the stretching process, and state the natural/independent variables that are measured/controlled at the system boundary. The system is the protein, and the variables that are measured/controlled at the system boundary are T , P , N , and l . b. (5 points) Write the fundamental energy equation corresponding to this system. The fundamental energy equation is: dU = TdS − PdV + μ j dN j j = 1 M ∑ + fdl c. (5 points) Given the natural variables from part a., write the fundamental equation corresponding to these variables. We use a Legendre transform to define: ξ = U − TS + PV . Then, d ξ = d U − TS + PV ( ) = dU − d TS ( ) + d PV ( ) = TdS − PdV + μ j dN j + fdl j = 1 M ∑ − TdS − SdT + PdV + VdP = − SdT + VdP + μ j dN j + fdl j = 1 M ∑ Incidentally, ξ = G . d. (10 points) Derive a Maxwell relation for calculating the change in entropy in the protein as a function of its length, in terms of measurable properties of the system. Use this to calculate the total entropy change during the stretching process. We want to find ∂ S ∂ l T , P , N . We note that S = − ∂ G ∂ T P , N , l . ChE 359 SP2010 – Exam #1 Page 2 of 4 Thus, we set up an Euler relation to get the Maxwell relation: ∂ S ∂ l T , P , N = − ∂ ∂ l ∂ G ∂ T P , N , l = − ∂...
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This note was uploaded on 02/09/2011 for the course CHE 359 taught by Professor Cynthialo during the Spring '10 term at Washington University in St. Louis.

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che359_sp10_exam1_soln - Page 1 of 4 ChE 359 Exam#1...

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