Chemistry 21b Problem set # 7 Out: 23Feb2018 Due: 02Mar2018

Transcription

1 Chemistry 21b Problem set # 7 Out: 23Feb2018 Due: 02Mar These are a suite of NMR or combined IR/NMR spectroscopy problems. From the spectra and information below, derive the molecular structure. Explain your reasoning. a). The formula for this compound is C 6 H 12 O 2. The IR spectrum is obtained on a neat liquid sample. What is the structure of this compound?

2 b). The compound whose IR and NMR spectra are shown below has a parent ion peak at 60 amu and a strong fragment at 31 amu. There is no UV absorption above 205 nm.

3 c). There is no UV absorption by this compound at wavelengths beyond the 205 nm cutoff in 95% ethanol. The IR spectrum is obtained on a neat liquid. The parent mass peak is at 74 amu, with a range of fragments from amu (and near 45 amu as well). From the spectra below, determine the structure.

4 d). The following compound, with the formula C 4 H 8 O 2, is an ester. Give its chemical structure and assign the chemical shifts to the appropriate proton(s). e). The following monosubstitued aromatic compound has the chemical formula C 9 H 12. Give its structure and assign the chemical shift values.

5 f). This compound has the chemical formula C 3 H 7 NO 2, and has two intense vibrational bands at 1550 and 1360 cm 1. Assign the chemical shifts below and the structure. g). And last but not least... This aromatic compound shows weak infrared absorption at about 3000, 2850, and 2750 cm 1 with strong absorption bands at 1680, 1260, 1030, and 840 cm 1. The fine structure in the NMR resonances of the aromatic protons can be used to assign the substitution structure on the benzene ring (see Section 6.3 of the on-line NMR supplementary notes from Week #7). What is the structure of this compound, and which protons produce the NMR chemical shifts in the spectrum below?

6 2. Here you will work out a couple of key relationships that are related to what is called the density matrix (in the Heisenberg formulation of quantum mechanics). Specifically, the density of a pure quantum state is defined as ρ ψ >< ψ Suppose we are in a complete, orthonormal basis set n >, such that the ket may be written as ψ > = n c n n > a. Write out theexpressionfor ρusingtheexpandedbraandket in the n>basis, andthus define the matrix elements ρ nm <n ρ m> in this basis set. From this, and the definition that <A> <ψ A ψ>, derive an expression for the expectation value of an operator A that involves the density matrix. b. Now, let s think about recasting the time dependent Schrödinger equation using the density matrix. Start by writing down and expression for (d/dt)ρ in terms of ψ> and <ψ, there should be two terms. And, combine this with the TSE, or d dt ψ > = ī h H ψ > to yield a compact expression for (d/dt)ρ (that should involve a commutator with H). Why do we care? This compact expression is called the Liouville-Von Neumann equation, and it can be shown to also be true for statistical ensembles. That is, if P k is the probability of a system being in a pure state ψ k >, then the density matrix of a statistical average can be written as ρ = k P k ψ k >< ψ k with P k 0 and k P k = 1 (for normalization). c. Finally, think about a two level system with basis functions 1> and 2>. What is the density matrix of a pure state that is the coherent superposition of the two basis functions, thatis ψ>= 1/ 2( 1> + 2>)? And,whatisthedensitymatrixforastatistical ensemble/average of both states where P 1 = P 2 = 0.5? Is there any wavefunction that can generate this statistical average?

7 3. Consider a system of N particles with mass m in a 3D box of volume V = L x L y L z, whose energy levels are given by: ǫ trans = ǫ x + ǫ y + ǫ z = h2 a 2 8mL 2 x + h2 b 2 8mL 2 y + h2 c 2 8mL 2 z, where a, b, and c are the particle-in-a-box quantum numbers (a,b,c = 1,2,3,...) for x, y, and z, respectively. A piston of area A = L x L y moves along the z direction to compress or expand the gas by changing the box length by dz. In the absence of heat flow, the reversible work dw done with this piston is simply dw = PdV = n j dǫ j, j where n j is the occupation number of state j = (a,b,c). By considering the change in each ǫ j due to dz, show that P = 2N < ǫ trans > 3V for any arbitrary 3D box; and show that this expression provides (yet another) route to the equation ofstateofaperfectgaswithnointernaldegrees offreedom. Theterm< ǫ trans >isthetranslational energy as averaged over the ensemble, not that for an individual particle (p. 246 of the class notes). Responsible TA for Problem Set #7: Griffin Mead (gmead@caltech.edu)