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(5-15)
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where 0 is the degeneracy of the reference energy state
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gibbs energy and the Biophysical Partition Function
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When applying statistical mechanics in biophysics, we are typically interested in conformational transitions in large biomolecules (such as DNA, proteins, and lipids) or binding interactions of biological significance In such cases the relevant energy change is the Gibbs energy change Other energy changes, for example,
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those between vibrational states, are very small relatively to the Gibbs energy change of binding or of a conformational transition Therefore the much smaller energy changes are negligible and can be ignored Practically speaking, they contribute only to the degeneracy of a particular Gibbs energy level Therefore it is most common in biophysical statistical mechanics to write the partition function as follows: Z = 1+
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Gi kT
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where DGi represents the Gibbs energy change in going from the reference state to the ith state of the system
chapter 5 s tat i s t i c a l M e c h a n i c s
Quiz
Refer to the text in this chapter if necessary Answers are in the back of the book 1 Which of the following statements about statistical mechanics are true S1 Statistical mechanics is the branch of mechanics that focuses on statistics S2 Statistical mechanics provides a molecular interpretation of a system S3 Statistical mechanics provides the mechanical tools necessary to measure statistical properties of molecules S4 Statistical mechanics relates statistical averages of particles (atoms, molecules, or residues) to the overall thermodynamic measurements of a system A S1 and S2 B s3 and s4 C S1 and S3 D s2 and s4 2 How many ways are there to distribute 8 units of energy among five molecules (Hint: Reread the section Finding Distributions) A 1 B 3 C 7 D 35 3 Given the distribution one molecule with 4 units of energy and four molecules with 1 unit of energy each, how many ways are there to arrange the molecules in the same distribution a 5 B 10 C 15 D 20 4 Given the distribution one molecule with 3 units of energy, one molecule with 2 units of energy, and three molecules with 1 unit of energy each, how many ways are there to arrange the molecules in the same distribution a 5 B 10 C 15 D 20 5 What is the probability of finding the system in each distribution of the three distributions, a, b, and c, shown in Fig 5-4 A 4%, 12%, and 4% B 20%, 60%, and 20% C 10%, 80%, and 10% D 2%, 6%, and 2%
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6 Given a system with a certain total amount of energy and a certain total number of molecules, the system will be most likely to distribute its energy among its molecules according to a which distribution has the most number of permutations B which energy has the most number of distributions c which molecule has the most energy D which distribution has the most number of molecules 7 Suppose we have a particular distribution of energy among a set of molecules For convenience we will call it distribution X How do we calculate the probability of distribution X relative to all possible distributions of the energy for this set of molecules a number of permutations for distribution X divided by total number of distributions B number of distributions for distribution X divided by the total number of distributions c number of distributions for distribution X divided by the total number of permutations for all possible distributions D number of permutations for distribution X divided by the total number of permutations for all possible distributions 8 The Boltzmann distribution is a the most probable distribution for any significant number of molecules B not the only distribution that occurs c so much higher in probability than all other distributions, that for all practical purposes, with any significant number of molecules it can be considered to be the only distribution that occurs D all of the above 9 A biophysicist wants to use a two-state approximation to model the conformational state of a solution of lipids That is, we assume that the lipids can exist in only two energy states Assuming a Boltzmann distribution and using E1 = 25 3 10 20 J and E2 = 30 3 10 20 J, what is the fraction of lipids in each energy level at 295 K a F1 = 773% and F2 = 227% B F1 = 223% and F2 = 777% c F1 = 763% and F2 = 237% D F1 = 890% and F2 = 110% 10 When an energy level has multiple states with the same amount of energy, then that energy level is said to be a diminished B decadent c degenerate D depraved
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