Chapter 19, Chemical Kinetics & Transition States Video Solutions, Molecular Driving Forces

Problem 1

Isotope substitution experiments can sometimes be used to determine whether hydrogens are cleared from molecules through mechanisms that involve tunneling. To test this, two isotopes are substituted for hydrogen: (1) deuterium (D, mass $=2$ ) is substituted and the ratio of rate coefficients $k_{H} / k_{D}$ is measured, and (2) tritium (T, mass $=3$ ) is substituted and $k_{H} / k_{T}$ is measured.
(a) Using the isotope substitution model in this chapter, show that
$$
\frac{k_{H}}{k_{T}}=\left(\frac{k_{H}}{k_{D}}\right)^{\alpha} .
$$
(b) Compute the numerical value of $\alpha$.

Lottie Adams

Numerade Educator

03:17

Problem 2

Relating stability to activation barriers. Using the Evans-Polanyi model, with $r_{1}=5, r_{2}=15, m_{1}=1$, and $m_{2}=-2$ :
(a) Compute the activation barriers $E_{a}$ for three systems having product stabilities $\Delta G=-2 \mathrm{kcal} \mathrm{mol}^{-1}$, $-5 \mathrm{kcal} \mathrm{mol}^{-1}$, and $-8 \mathrm{kcal} \mathrm{mol}^{-1}$.
(b) Plot the three points $E_{a}$ versus $\Delta G$, to show how the activation barrier is related to stability.

Chai Santi

Numerade Educator

03:24

Problem 3

Reduced masses. Equation (19.31) gives the reduced masses for $\mathrm{C}-\mathrm{H}$ and $\mathrm{C}-\mathrm{D}$ bonds as $\mu_{\mathrm{CH}} \approx m_{\mathrm{H}}$ and $\mu_{\mathrm{CD}} \approx$ $2 m_{\mathrm{H}}$. The approximation is based on assuming the mass of carbon is much greater than of $\mathrm{H}$ or D. Give the more correct values of these reduced masses if you don't make this assumption.

George Bennett

Numerade Educator

07:37

Problem 4

Classical collision theory. According to the kinetic theory of gases, the reaction rate $k_{2}$ of a sphere of radius $r_{A}$ with another sphere of radius $r_{B}$ is
$$
k_{2}=\pi R^{2}\left(\frac{8 k T}{\pi \mu_{A B}}\right)^{1 / 2} e^{-\Delta \varepsilon_{0}^{\ddagger} / k T},
$$
where $R=r_{A}+r_{B}$ is the distance of closest approach, $\mu_{A B}$ is the reduced mass of the two spheres, and $\Delta \varepsilon_{0}^{7}$ is the activation energy. Derive this from transition state theory.

Sowmya Ragothaman

Numerade Educator

00:35

Problem 5

(a) Show that the pressure dependence of the rate constant $k$ for the reaction
$$
A \stackrel{k_{f}}{\longrightarrow} B
$$
is proportional to an activation volume $v^{\neq}$:
$$
\left(\frac{\partial \ln k_{f}}{\partial p}\right)=-\frac{\left(v^{\neq}-v_{A}\right)}{k T} .
$$
(b) Show that the expression in (a) is consistent with Equation (13.46), $K=k_{f} / k_{r}$, where $k_{r}$ is the rate of the reverse reaction.

Ronald Prasad

Numerade Educator

01:24

Problem 6

Relating the Arrhenius and activated-state parameters. Derive the relationship of the activation parameter $\Delta H^{*}$ in Equation (19.26) to the Arrhenius activation energy $E_{a}$ in Equation (19.11) for a gas-phase reaction.

Hunza Gilgit

Numerade Educator

01:42

Problem 7

Enzymes accelerate chemical reactions. Figure $19.22$ shows an Arrhenius plot for the uncatalyzed reaction of 1-methylorotic acid (OMP) [7].
(a) Estimate $\Delta H^{\ddagger}$ from the figure.
(b) Estimate $\Delta S^{\ddagger}$ at $T=300 \mathrm{~K}$.
(c) At $T=25^{\circ} \mathrm{C}$, the enzyme OMP decarboxylase accelerates this reaction $1.4 \times 10^{17}$-fold. How fast is the catalyzed reaction at $25^{\circ} \mathrm{C}$ ?
(d) What is the binding constant of the enzyme to the transition state of the reaction at $T=300 \mathrm{~K}$ ?

Dominador Tan

Numerade Educator

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Problem 8

Rate increase with temperature. A rule of thumb used to be that chemical reaction rates would roughly double for a 10-degree increase in temperature, say from $T_{1}=300 \mathrm{~K}$ to $T_{2}=310 \mathrm{~K}$. For what activation energy $E_{a}$ would this be exactly correct?

David Collins

Numerade Educator

04:11

Problem 9

Catalytic rate enhancement. The reaction rate of an uncatalyzed reaction is $k_{0}=10^{3} \mathrm{M}^{-1} \mathrm{~s}^{-1}$ at $T=300 \mathrm{~K}$. A catalyst $C$ binds to the transition state with free energy $\Delta G=-5 \mathrm{kcal} \mathrm{mol}^{-1}$. What is the rate $k_{0}$ of the catalyzed reaction at $T=300 \mathrm{~K}$ ?

Niamat Khuda

Numerade Educator

04:11

Problem 10

The reaction rate of an uncatalyzed reaction is $k_{0}=10^{3} \mathrm{M}^{-1} \mathrm{~s}^{-1}$ at $T=300 \mathrm{~K}$. A catalyst $C$ binds to the transition state with free energy $\Delta G=-5 \mathrm{kcal} \mathrm{mol}^{-1}$. What is the rate $k_{0}$ of the catalyzed reaction at $T=300 \mathrm{~K}$ ?

Niamat Khuda

Numerade Educator

01:47

Problem 11

Drug $A$ is an uncharged molecule containing a three-ring aromatic chromophore linked to three amino acids: valine, $N$ methylvaline and proline. This drug binds to doublestranded DNA, with the planar ring system intercalating between base pairs and the peptide portion lying in the minor groove.

Ramesh Singh

Numerade Educator

01:13

Problem 12

(a) You have a reaction in which $A$ converts to $B$ through two steps in series:
$$
A \stackrel{k_{1}}{\longrightarrow} I \stackrel{k_{2}}{\longrightarrow} B,
$$
where the temperature dependences of the steps are given by the Arrhenius law:
$$
\begin{aligned}
&k_{1}=a e^{-E_{1} / R T}, \\
&k_{2}=a e^{-E_{2} / R T} .
\end{aligned}
$$
Derive an expression for the total rate $k_{\text {tot }}$, from $A$ to $B$ as a function of a, $E_{1}, E_{2}$, and T. (Hint: the time from $A$ to $B$ is the sum of times, from $A$ to $I$ and from $I$ to $B$.)
(b) Now consider instead two steps in parallel:
$$
A \stackrel{k_{1}}{\longrightarrow} B, \quad A \stackrel{k_{2}}{\longrightarrow} B
$$
Using the Arrhenius equations from (a) for the individual steps, derive the temperature dependence for $k$ in this case. (Hint: now the rates add.)

David Collins

Numerade Educator

02:13

Problem 13

The frequency that crickets chirp increases with temperature. You can calculate the temperature in degrees Celsius by adding 4 to the number of chirps you hear from a cricket in $8 \mathrm{~s}$. In this way, you can use a cricket as a thermometer.
(a) What four temperatures would you have measured if the number of chirps in 8 seconds were $16,21,26$, and 31 ?
(b) From this data, make an Arrhenius plot and calculate the activation energy in kcal $\mathrm{mol}^{-1}$. This will tell you about a rate-limiting biochemical reaction that underlies cricket chirping.
(c) You observe that crickets outside a fast-food restaurant chirp faster than their cousins in the wild. The fast-food crickets chirp 30 times in $8 \mathrm{~s}$ at $25^{\circ} \mathrm{C}$ and 40 times in $8 \mathrm{~s}$ at $35^{\circ} \mathrm{C}$. By how much did the fast-food additive reduce the activation barrier for cricket chirping?
(d) If the equilibrium enthalpy for the chirping reaction, $\Delta h^{\circ}$, is $10 \mathrm{kcal} \mathrm{mol}^{-1}$, what is the activation energy for the reverse chirp reaction for the normal crickets?

James Erikson

Numerade Educator

00:48

Problem 14

Huntington's disease is an example of a polyglutamine disease. The severity of the disease correlates with the degree of polyglutamine fibril formation. The age of onset decreases exponentially with increasing length of the polyglutamine chains (see the curve in Figure 19.26). Fibrils form quickly after an initial nucleation event, the binding of two polyglutamine chains. Assume dimerization is the ratelimiting step in the onset of the disease. Our aim is to interpret this data with a simple dimerization model.
(a) Use the lattice model shown in Figure $19.27 .$ Define $w_{s s}, w_{s g}$, and $w_{g g}$ to be the solvent-solvent, solvent-glutamine, and glutamine-glutamine interaction energies. If each glutamine chain is of length $L$, fill in Table $19.4$ for the pairwise interactions that are broken and formed when two chains dimerize.
(b) Write an expression for the dimerization equilibrium constant in terms of the interaction energies.
(c) Use the expression from answering (b) to explain the exponential form of Figure $19.26 .$
(d) Estimate the polymer-solvent interaction energy $\chi_{s g}$ from Figure $19.26 .$

Sana Riaz

Numerade Educator