Chemistry 2710 Spring 2000 Practice Problems

1.

The vibrational frequency of carbon monoxide is $6.3\times 10^{13}\,\mathrm{Hz}$.

(a)

Calculate the difference in energy between any two adjacent vibrational energy levels.

(b)

Calculate the ratio of the populations of adjacent vibrational levels at 298K.  

(c)

Because the vibrational spacing is (roughly) constant, we can calculate the probability that a molecule occupies the lowest energy level directly. Suppose that the population ratio calculated in question 1b is f. Then, N₁ = fN₀, N₂ = fN₁ = f²N₀, etc. The total number of molecules in all energy levels is therefore

$$N = N_0 + fN_0 + f^2N_0 + \ldots = N_0\sum_{i=0}^\infty f^i.$$

The infinite sum above can be shown to have the value

$$\sum_{i=0}^\infty f^i = \frac{1}{1-f}.$$

Therefore, the probability that any given molecule occupies the lowest energy level is

$$\mathrm{Pr}(n=0) = \frac{N_0}{N} = 1-f.$$

Calculate the probability that a carbon monoxide molecule is in the lowest vibrational energy level at 298K.

(d)

Use the equipartition principle to estimate the average total energy of a carbon monoxide molecule. The experimental result is $(3.455\times 10^{-23}\,\mathrm{J/K})T$ to T near 298K. How does the equipartition value compare?

2.

In the Diels-Alder reaction of ethene with 1,3-butadiene to form cyclohexene in the gas phase, the activation energy is 115kJ/mol. The following thermochemical data for the reactants and products are available:

Species	$\Delta\bar{E}^\circ_f$ (kJ/mol)
C2H4(g) (ethene)	54.95
C4H6(g) (1,3-butadiene)	115.4
C6H10(g) (cyclohexene)	14.24

(a)

Sketch the energy profile of the reaction. Clearly label the $\Delta\bar{E}$ for the reaction as well as the activation energies for the forward and reverse reactions.

(b)

Calculate the activation energy for the reverse reaction ( $\mathrm{cyclohexene}\rightarrow\mathrm{ethene}+\mathrm{butadiene}$).

(c)

Calculate the enthalpies of activation for the forward and reverse reactions at 298K.

3.

The enthalpies of formation of ethane ( C₂H₆) and of methyl radicals ( CH₃) are, respectively, -83.85 and 145.69kJ/mol at 298.15K. Calculate the activation energy for the dissociation of ethane into methyl radicals. Hint: Sketch the expected energy profile for this reaction.

4.

Azomethane ( CH₃N₂CH₃) decomposes to methyl radicals and nitrogen at elevated temperatures in the gas phase by a first-order process. The rate constant depends on temperature as follows:

T (K)	523	541	560	576	593
k ( s-1)	$1.8\times 10^{-6}$	$1.5\times 10^{-5}$	$6.0\times 10^{-5}$	$1.6\times 10^{-4}$	$9.5\times 10^{-4}$

(a)

Calculate the activation energy and preexponential factor for this reaction.

(b)

Calculate the entropy of activation at 550K. Based on your calculation, comment on the nature of the transition state for this reaction.

5.

For first-order reactions in solution near room temperature, at what approximate value of $k_\infty$ would we have $\Delta\bar{S}^\ddagger = 0$? On a molecular level, what is the significance of such a value of the entropy of activation?