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Questions found on previous Third Hour Exams; the answers are in curly brackets.

D stands for delta; for example, DH represents delta H.

Chemistry 142 (Relevant questions from the third hour exam, Spring 1992)

3(21) - The reaction: H2O2(g) = H2(g) + O2(g)

is first order at high pressures, and second order at low pressures. Just for grins, assume

kR = 0.010, where the units are moles/L, (raised to the proper power) and seconds.

a) Calculate the concentration of H2O2 after one hour, if the
initial concentration is 1.00M, and we are at the high pressure
limit. **{2.3x10-16 M}**

b) Now, calculate the concentration of H2O2 after one hour,
starting with an initial concentration of 1.00 M, and we are in the
low pressure limit. **{0.027 M}**

c) Write a possible mechanism for the reaction that would account for the switch in orders. {The Lindemann-Hinshelwood mechanism, discussed in lecture}

4(20) - Consider the reaction: 2 O3 = 3 O2

A proposed mechanism is

O3 = O2 + O

O + O3 = 2 O2

a) Using the steady state method, derive an expression for the
rate equation, in terms of the concentrations of O2 and O3, assuming
that the rate of the reaction is the rate of the last reaction.
**{rate = k2[O3]2/[k-1[O2] + k2[O3]]}**

b) Under what circumstances (in terms of k's and concentrations)
would the first reaction be the RDS? (Use the Law of Connick and an
inequality that simplifies the expression in a)). **{k2[O3] >>
k-1[O2]}**

5(12) - A theoretical expression for a rate constant is

kr = CT2e-E/RT, where C is independent of temperature.

Show that E = Ea - 2RT. (I'll sell you the general equation for Ea if you need it.)

6(25) - a) Calculate the average momentum in the x direction for a 1-d pib. {zero}

b) Is psi an eigenfunction of ^px? Show why. **{no}**

c) On our last problem set, we showed that the probability of a
ground state particle being in the left third of the box is 0.196.
Without doing any calculus, use the symmetry of the ground state
wavefunction (and the normalization condition) to calculate the
probability of the particle being in the middle third of the box.
**{0.608}**

(Third Exam, 1993)

1(25) - The reaction 2 A = B involves a catalyst, C. The proposed mechanism is

k1

A + C = AC

k-1

k2

AC + A = B

a) Derive an expression for the steady state concentration of the intermediate, AC.

**{[AC] = k1[A][C]/{k-1 + k2[A]}}**

b) If the rate of the reaction is dB/dt, write the equation for
the rate equation, in terms of A and C (but not AC). **{rate =
k1k2[A]2[C]/(k-1 + k2[A])}**

c) We observe that the reaction is initially first order in A, but later becomes second order in A. (It is first order in C throughout.)

i) What is the initial RDS? Justify, using Connick's Law. **{A +
C = AC}**

ii) What is the RDS later on? Justify. **{A + AC = B}**

d) Why does the order change with time (i.e., first order at the beginning, second order near the end) the way it does?

2(25) - We want to synthesize an organic compound, G (for Good). However, we're concerned about a side reaction that produces the despised Intractable Tar, B (for Bad, obviously). If the side reaction is favored by having a larger DS* (DS(B) - DS(G) = 10 J/mol-K), but is inhibited by its higher activation energy (Ea(B) - Ea(G) = 10 kJ/mol),

a) Calculate the temperature at which at least 95% of the product will be the desired one (G).

**{290 K}**

b) In order to reduce the chance of contamination even further,
should we run the reaction at a higher or lower temperature? Explain,
using physical, not organic, ideas. **{lower T}**

3(25) - a) Identify the following hydrogenic orbitals (3dxy, 4px, etc.), stating your reasoning.

i) psi = Ne^{-Zr/ao} **{1s}**

ii) psi = Ne^{-Zr/4ao}(Zr/ao)^{2}(6 -
Zr/ao)sin^{2}(theta)cos(phi)sin(phi) **{4dxy}**

b) Find the normalization constant for the first wavefunction.
**{(Z ^{3}/(pi)a_{o}^{3})^{1/2}**

(integrale^{-ax}x^{n}dx = n!/a^{n-1} ; x =
rsin(theta)cos(phi)

y = rsin(theta)sin(phi) z = rcos(theta)

4(25) - (No calculus is required for parts a,b and c.) Consider a particle in a box of length a. We know that the average value of the position is x = a/2. We also know that

E_{n }= n^{2}h^{2}/8ma^{2} =
p_{x}^{2}/2m,

where px is the magnitude of the momentum in the x direction.

a) What is the uncertainty of x? That is, if x = x +/- delta(x), what is delta(x)?

b) We showed that the average value of the momentum was zero; use the energy equation to calculate the magnitude of px, and determine the uncertainty in momentum.

c) Is your product delta(x).delta(px) consistent with the Heisenberg Uncertainty Principle, which states that delta(x).delta(px)> h/4(pi)?

d) Now, use The Calculus. Using (psi) = (30/a5)2x(a - x),
calculate p_{x}. ((psi) is normalized.) **{0}**

p^x = -i(h/2(pi))(d/dx); (psi) = sqrt[(2/a)]sin(n(pi)x/a)

(integral)sin(ax)cos(ax)dx = (1/2a)sin^{2}(ax) + C

Third Hour Exam Spring, 1994

1(26) - A possible mechanism for the enzyme catalyzed reaction

A + B = P is

A + E = EA (with forward rate constant k1, and reverse rate constant, k-1)

B + EA = P + E (with forward rate constant k2, and no reverse rate constant)

(8) a) Use the steady state approximation for [EA] to show that

[EA] = k1[A][E]/{k-1 + k2[B]}

(6) b) If the overall rate of the reaction is d[P]/dt, write a general equation for the rate of the reaction, in terms of the concentrations of A, B, and E (but not EA).

**{rate = k2k1[B][A][E]/(k-1 + k2[B])}**

(6) c) Under what conditions would the reaction be zeroth order with respect to B?

**{k2[B] much greater than k-1}**

(6) d) Which reaction above would be the Rate Determining Step
under the conditions in (c)? Explain, using the Law of Connick. **{A
+ E = EA}**

2(16) - A particular reaction has an Arrhenius Activation Energy of 85.2 kJ/mol.

(10) a) What would be the percent increase in the rate constant if
the temperature were increased from 298 K to 310 K? **{278%}**

(6) b) If the Eyring equation for the rate constant is
k_{R} = [kT/h]e^{DS/R}e^{-DH/RT},

Calculate the value of DH at 310 K. (If you need the general
equation for Ea, just ask.) **{82.6 kJ/mol}**

4(33) - A reasonable approximate wavefunction for helium is

psi = psi_{1s}(1)*psi_{1s}(2), where
psi_{1s}(1) =
(X^{3}/(pi)a_{o}^{3})^{1/2}e-Xr_{1}/a_{o},
and psi_{1s} is normalized.

(6) a) Show, without doing any hard calculus, that psi is normalized.

(Hint: d(tau) = d(tau)_{1}*d(tau)_{2})

(8) c) Show what value of X would lead to values of _T and _V that are consistent with the Virial Theorem.

(10) d) Use psi_{1s}(1) to calculate the average distance
of a 1s electron from the nucleus of a helium atom.
(integral)x^{n}e^{-ax}dx = n!/an+1, where the
integral is from 0 to infinity. (As

usual, if you need any other integrals, ask for them. After all, I am your servant.)

Chemistry 142 Third Hour Exam Spring, 1995

1(22) - Consider the enzyme catalyzed reaction: 2 S = P. If the proposed mechanism is

S + E = ES (forward rate constant k1, reverse rate constant k-1)

S + ES = E + P (where k2 is the rate constant.)

a) Using the steady state approximation for the intermediate, ES,
derive an expression for [ES] as a function of [S], [E] and rate
constants. **{[ES] = k1[E][S]/(k-1 + k2[S])}**

b) Assume that the rate of the last reaction is the overall rate of the reaction, and derive an equation for this rate, in terms of the concentrations of E and S, and rate constants.

c) If the rate is given by: rate = k[E][S]2/(1 + k'[S]), how could
we force the reaction to be second order in S? **{make [S] be very
small}**

d) Under the conditions in c), what would be the rate determining
step? Explain. **{last step}**

2(16) - A certain reaction may proceed by a direct path or a
catalyzed path. If DS_{path b} =

DS_{path a} a + 40 J/mol-K, and DH_{path b }=
DH_{path a} + 20 kJ/mol,

a) Which path is the catalyzed path? (a or b?) Explain. **{Path
a}**

b) Which path has the greater rate constant at 310 K, and by what
factor? (It is not necessary to calculate either kr!) **{Path a is
19 times faster.}**

3(20) - a) Physicists prefer to use the angular hydrogenic
wavefunction, PHI(phi) = Ne^{im(phi)}, where i =
(-1)^{1/2}, m is an integer, and 0 <phi<2(pi). Show
that N = 1/(2(pi))^{1/2}

b) If a normalized molecular orbital is(psi)_{mo} =
0.70(psi)_{1sA} - 0.90(psi)_{1sB}, and each of the 1s
atomic orbital wavefunctions is normalized, what is the numerical
value of the overlap integral,
*(integral)(psi)_{1sA}(psi)_{1sB}d(tau)? (Do not do
any calculus in part b) **{0.24}**

c) Which end, A or B, has the higher density if the molecular
orbital in b) is occupied? Explain.**{B}**

5(20) - For the following orbitals:

b) Plot the radial function of a 4s orbital vs. r. Indicate the number of nodes.(3 nodes)

c) if z = r cos(theta) and x = r sin(theta) cos(phi), write the equation for a 4dxz wavefunction

as completely as you can. {(psi) = Ne^{-Zr/4a}or2(a -
br)sin(theta)cos(phi)}

1. (35 pts) The acid catalyzed bromination of acetone

CH3COCH3 + Br2 = CH3COCH2Br + HBr

was studied using the initial rate method, and the following data were obtained at 25 oC:

Initial molarity of

Run Acetone H+ Br2 Initial rate (M/s)

1 2.00 0.100 0.0250 1.50 x 10-2

2 3.00 0.100 0.0250 2.25 x 10-2

3 2.00 0.175 0.0250 2.53 x 10-2

4 2.00 0.100 0.0150 1.40 x 10-2

(a) Determine the order of the reaction with respect to acetone,
H+ and Br2. **{1.00, 0.934, 0.135}**

(b) Calculate the rate constant for the reaction, including its
units. **{0.0750 L/mol-sec}**

(c) If 1.00 M acetone is mixed with 1.00 M H+ and 0.50 M Br2, how
long will it take for the concentration of acetone to fall to 0.75 M?
**{4.44 sec}**

(d) Propose a possible mechanism for the reaction, indicating which is the Rate Determining Step.

(e) When the first run is repeated at 35 oC, the rate increases to
3.15 x 10-2 M/s. What is the activation energy for the reaction?
**{56.6 kJ/mol}**

2. (30 pts) The reaction 2 NO (g) + O2 (g) = 2 NO2 (g)

was once thought to be an elementary third order reaction. More recently it has been proposed that the reaction proceeds by the following mechanism:

(i) 2 NO = N_{2}O_{2 }(rate constant =
k_{1})

(ii) N_{2}O_{2} = 2 NO (rate constant =
k_{2})

(iii) N_{2}O_{2} + O_{2 }= 2
NO_{2} (rate constant = k_{3})

(a) Derive an equation that gives the steady state concentration
of N_{2}O_{2}.

(b) Use the steady-state approximation to derive a rate expression
for the net reaction (e. g., -d[O_{2}]/dt). Under what
assumption will your expression look third order?

(c) How could you fiddle with the concentration of O_{2}
to test this proposed mechanism? What results, i.e., what change of
order with concentration, would support the mechanism?

3. (15 points) As you are undoubtedly aware, both Bohr and Schroedinger agreed that the energy of any one-electron atom or ion is

E = -Z2B/n2, where B = 2.18x10^{-18} Joules.

a) If a ground state 3Li2+ ion absorbs a photon with an
energy of 1.744x10^{-17} Joules, calculate the final
principal quantum number, n. **{n = 3}**

b) Use the Virial Theorem to calculate the potential energy of a
3p electron on a Li2+ ion. **{-4.36x10 ^{-18} J}**

4.(20 points) As you know, for a particle in a one dimensional box, psi = (2/a)2sin(n(pi)x/a)

(a) Show how one would calculate the average value of the position, _x, of a particle in a one-dimensional pib. (You don't have to do the calculation; just set it up.)

(b) For a one-dimensional pib, show whether or not (psi)is an
eigenfunction of the kinetic energy operator,
-(h^{2}/2m)d^{2}/dx^{2}.