3.012 Quiz 4 3.0123.012 Quiz 4 3.012 12.15.04 YOUR NAME: ________________________________ Fall 2004 100 points total Time yourself carefully – do not spend most of your time on a single question. Remember, you can take up to 3 hours. Give as much written explanation as possible of your reasoning, and write clearly and legibly. Bonding. (50 points) 1. (10 points) What are the quantum numbers that describe the eigenfunctions of an electron in a hydrogen atom ? How are these quantum numbers called, what do they refer to, and what are the possible values that they can take ? Why do we use these quantum numbers in the case of a hydrogen atom, or in general for the case of a central potential ? 3.012 Quiz 4 1 of 5 12/14/042. (10 points) What are the orbitals involved in the bonding between the two carbon atoms in a molecule of ethane (C2H6), of ethene (also known as ethylene, C2H4), and of ethyne (also known as acetylene, C2H2) ? Which molecule will have the shortest distance between the carbons ? And which molecule will require more energy to break the bond between the carbons ? 3.012 Quiz 4 2 of 5 12/14/043. (10 points) What are the molecular orbitals of a homonuclear diatomic molecule ? Choose one among the second row elements (i.e. Li2 or Be2 or B2 or C2 or N2 or O2 or F2 or Ne2). What would be the Hamiltonian for this molecule, in atomic units ? Also, plot the total energy of the molecule as a function of the distance between the nuclei: what determines the equilibrium distance ? 3.012 Quiz 4 3 of 5 12/14/044. (10 points) Describe the method of “linear combinations of atomic orbitals” and its application to the study of bonding in molecules. What is its relation to the variational principle ? Actually, what is the variational principle ? 3.012 Quiz 4 4 of 5 12/14/045. (10 points) What is the expectation value for the distance from the nucleus of an electron in a hydrogen atom, when it is in a 1s state ? (Note: in atomic units, and using spherical coordinates, 11ersψπ−= ; also, 101!narnre dr na∞−+=∫ , where n is an integer, a is a real positive number, and n! is n factorial, i.e. n*(n-1)*(n-2)*…*2*1, that is to say that 1!=1, 2!=2, 3!=6, 4!=24, 5!=120 and so on…). 3.012 Quiz 4 5 of 5 12/14/043.012 Quiz 4 6 12/15/04Thermodynamics. (50 points)6. Shown below on the left is a binary phase diagram, and on the right the same diagram withannotation. Use these diagrams to answer the questions below:XBTLα + Lα + βαββ + LXBTLα + Lα + βαββ + L01T1T2X’X’’a. (3 points) On the left-hand diagram, mark the invariant point(s) and write the equilibriumthat exists at each in the space below.b. (4 points) For temperature T2, identify the phases present, the composition of thephases, and the phase fraction of each phase present at the composition X’ and at thecomposition X’’. (Identify additional points on the diagram as needed.)3.012 Quiz 4 7 12/15/04c. (4 points) What is the difference between the phase rule for a single component T/Pdiagram and the phase rule for a binary T/XB phase diagram? Explain your answer with1-2 sentences.d. (4 points) In the space below, sketch a reasonable set of molar free energy vs.composition curves for each phase of the binary system at the temperature marked T1above.3.012 Quiz 4 9 12/15/048. A system of N distinguishable molecules that each have 3 possible energy states (with energies– ε, 0, and + ε) is equilibrated at a temperature T.a. (6 points) What is the partition function of the entire system of N molecules?b. (6 points) Show that the ratio € pεp−ε, the probability to find one molecule of the system inthe E = ε state over the probability of one molecule residing in the E = - ε state can bedetermined without knowledge of the partition function, and calculate this ratio.c. (7 points) Show that the free energy of this system becomes a linear function of T at veryhigh temperatures.3.012 Quiz 4 10 12/15/049. (6 points) In our work on binary solution phase behavior, we showed that some degree ofmixing (homogeneous solution formation on addition of a small amount of the component B topure A and vice versa) is always favored if the components form a regular solution- even thoughthe two components may be very chemically incompatible (i.e., they have a very large interactionparameter Ω). Mixing in this situation is driven by a favorable entropy change. Usingarguments based on the statistical mechanical definition of entropy, explain in a few sentenceswhy entropy favors mixing. (You may invoke a lattice model description of the binary solution tohelp with your
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