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373 Hw3 crib

# 373 Hw3 crib - and so the average kinetic energy is once...

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UPC Crib 3 10/14/2010 1a. Following the derivation on page 32, b. If the degeneracy ratio is 1, then taking the natural log of the following equations yields the energy difference, c. Solving for the degeneracy ratio given , There are about 1500 unfolded conformations. This is a realistic number since there are many unfolded conformations but only one which is folded. 2a. The Hamiltonian is the sum of the kinetic and potential energies, . A particle confined to one dimension has one translational degree of freedom, and with the provided quadratic potential the Hamiltonian is

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b. Since the kinetic energy gives rise to a single quadratic term in the Hamiltonian, the average kinetic energy is c. The Virial Theorem relates the ratio to the kinetic and potential energies for any system whose potential energy has the following form . For a harmonic potential n = 2, thus Using the information from part b, we can solve for the average potential energy to get d. The functional form of the kinetic energy is again a quadratic function of velocity

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Unformatted text preview: and so the average kinetic energy is once again . We may now use the Virial Theorem (as in c) to determine the average potential energy of this system, for which n = 6, 3a. From eq. 1.31, The average speed is where we define and . The general solution to an integral of this form is For m = 3, Remembering that , the solution to the integral is The last step was performed recalling and . b. The most probable velocity is reached at the maximum of the probability distribution. Taking the derivative of P(v) with respect to velocity and setting the value equal to zero will give the maximum velocity. Recalling the values of the constants c 1 and c 2 , after factoring out c 1 e c 2 v 2 2 v the above result implies that 1 c 2 v 2 = when v = v max and so c. The required velocities are obtained using the molecular weights of N 2 (28 g/mol) and H 2 (2 g/mol), as well as using the fact that in the SI unit system 1 J = 1 kg (m/s) 2 ....
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