(c) (8 pts) Modify the above chemical potential expression (c) into your (c), so that (a')(b)(c') make dS an exact differential. Obtain the entropy expression S(U,V,n) from (a')(b)(c').

(A) A) is the 1H NMR spectrum of 3-hetanone. Hydrogens associated with each carbon are labeled in the molecular structure shown below. Assigned each group of peaks from left to right on the NMR spectrum with lower-case letters (a-f). (4%)

(B) Homonuclear decoupling is a technique where the sample to be analyzed is irradiated at a select frequency to eliminate the effect of coupling due to certain nuclei. In other words, the signal of a select nucleus can be selectively saturated, such that this nucleus can no longer couple to neighboring nuciei. This technique helps to simplify complicated peak splitting. Irradiation at which group of protons (a-f) would result in spectral simplification in the 2.3 - 2.6 ppm range as shown in B)? Use the same lower-case letter(s) notation in A). (3%)

(a) (8 pts) Integrate the first and the second laws Tds = dU + PdV - μdn, together with the above conditions (a)(b)(c), to obtain the entropy formula S(U, V,n), which is a function of U,V and n (and T, P, μ are all eliminated).

(b) (4 pts) Suppose that a slightly different gas system satisfies (a') U= nCTδ but not (a), where the constant δ  ≠1. Show that the conditions (a')(b)(C) imply that the entropy change dS is not an exact differential.

(a) (2 pts) Write down the rate laws for d A(t)]/dt and d[B(t)]/dt.

(b) (3 pts) Initially [A(t = 0)] = A0, and [B(t = O) = [C(t = 0)] = 0. Apply the steady state approximation on the intermediate B, to solve the product concentration [C(t)].

(c) (5 pts) Without using the steady state approximation, solve the concentra- tion [C(t)] exactly from the rate laws. What is the condition under which the steady state approximation applies?

(a) (2 pts) Show that the hamiltonian is hermitian.

(b) (A pts) Sol've the energy eigenvalues and their eigenvectors.

(c) (2 pts) Suppose that at t = O the proton is at the state Calculate the normalization constant N.