Question

A study of the rate of dimerization of \({{\bf{C}}_{\bf{4}}}{{\bf{H}}_{\bf{6}}}\) gave the data shown in:

\({\bf{2}}{{\bf{C}}_{\bf{4}}}{{\bf{H}}_{\bf{6}}} \to {{\bf{C}}_{\bf{8}}}{{\bf{H}}_{{\bf{12}}}}\)

1. Determine the average rate of dimerization between 0 s and 1600 s, and between 1600 s and 3200 s.
2. Estimate the instantaneous rate of dimerization at 3200 s from a graph of time versus (\({{\bf{C}}_{\bf{4}}}{{\bf{H}}_{\bf{6}}}\)). What are the units of this rate?

(c) Determine the average rate of formation of \({{\bf{C}}_{\bf{8}}}{{\bf{H}}_{{\bf{12}}}}\) at 1600 s and the instantaneous rate of formation at 3200 s from the rates found in parts (a) and (b).

Short answer

1. The average rate of dimerization between 0s and 1600s was found to be\({\bf{3}}{\bf{.1 \times 1}}{{\bf{0}}^{{\bf{ - 6}}}}\)M/s and rate of dimerization between 1600s and 3200s was found to be\({\bf{1}}{\bf{.04 \times 1}}{{\bf{0}}^{{\bf{ - 6}}}}\)M/s
2. The instantaneous rate of dimerization at 3200s is\({\bf{ - 7}}{\bf{.83 \times 1}}{{\bf{0}}^{{\bf{ - 7}}}}\)M/s and units of this rate is M/s
3. The average rate of formation of \({{\bf{C}}_{\bf{8}}}{{\bf{H}}_{{\bf{12}}}}\) at 3200s is \({\bf{1}}{\bf{.55 \times 1}}{{\bf{0}}^{{\bf{ - 6}}}}\) M/s

Step-by-step solution

Average rate of dimerization

Rate of dimerization= -\(\frac{{{\bf{\Delta (C}}{}_{\bf{4}}{\bf{H}}{}_{\bf{6}}{\bf{)}}}}{{{\bf{Vt}}}}\)

Rate of dimerization between 0s and 1600s

\(\begin{aligned}{}{\bf{ = }} - \frac{{{\bf{5}}{\bf{.04 \times 10}}{}^{{\bf{ - 3}}}{\bf{M}} - {\bf{1}}{\bf{.00 \times 10}}{}^{{\bf{ - 2}}}{\bf{M}}}}{{{\bf{1600s}} - {\bf{0s}}}}\\{\bf{ = 3}}{\bf{.10 \times 1}}{{\bf{0}}^{{\bf{ - 6}}}}{\bf{M/s}}\end{aligned}\)

Rate of dimerization between 1600s and 3200s

\(\begin{aligned}{}{\bf{ = }} - \frac{{{\bf{3}}{\bf{.37 \times 10}}{}^{{\bf{ - 3}}}{\bf{M}} - {\bf{5}}{\bf{.04 \times 10}}{}^{{\bf{ - 3}}}{\bf{M}}}}{{{\bf{3200s}} - {\bf{1600s}}}}\\{\bf{ = 1}}{\bf{.04 \times 1}}{{\bf{0}}^{{\bf{ - 6}}}}{\bf{M/s}}\end{aligned}\)

Instantaneous rate of dimerization      

Graph of time versus (\({{\bf{C}}_{\bf{4}}}{{\bf{H}}_{\bf{6}}}\))

Slope between 1600s ­­­­­­­­and 3200s

Use points (1600s,\({\bf{5}}{\bf{.04 \times 10}}{}^{{\bf{ - 3}}}\)M) and (3200s,\({\bf{3}}{\bf{.37 \times 10}}{}^{{\bf{ - 3}}}\)M)

\(\begin{aligned}{}{\bf{ = }}\frac{{{\bf{3}}{\bf{.37 \times 1}}{{\bf{0}}^{{\bf{ - 3}}}}{\bf{M - 5}}{\bf{.04 \times 10}}{}^{{\bf{ - 3}}}{\bf{M}}}}{{{\bf{3200s - 1600s}}}}\\{\bf{ = - }}\frac{{{\bf{0}}{\bf{.00167M}}}}{{{\bf{1600s}}}}\\{\bf{ = - 1}}{\bf{.04 \times 10}}{}^{{\bf{ - 6}}}{\bf{M/s}}\end{aligned}\)

Slope between 3200s ­­­­­­­­and 4800s

Use points (3200s,\({\bf{3}}{\bf{.37 \times 10}}{}^{{\bf{ - 3}}}\)M) and (4800s, \({\bf{2}}{\bf{.53 \times 10}}{}^{{\bf{ - 3}}}{\bf{M}}\))

\(\begin{aligned}{}{\bf{ = }}\frac{{{\bf{2}}{\bf{.53 \times 1}}{{\bf{0}}^{{\bf{ - 3}}}}{\bf{M - 3}}{\bf{.37 \times 10}}{}^{{\bf{ - 3}}}{\bf{M}}}}{{{\bf{4800s - 3200s}}}}\\{\bf{ = - }}\frac{{{\bf{0}}{\bf{.00084M}}}}{{{\bf{1600s}}}}{\bf{ = - 5}}{\bf{.25 \times 10}}{}^{{\bf{ - 7}}}{\bf{M/s}}\end{aligned}\)

Take two slope and find average of instantaneous rate of dimerization

\(\begin{aligned}{}{\bf{ = }}\frac{{{\bf{ - 1}}{\bf{.04 \times 10}}{}^{{\bf{ - 6}}}{\bf{ + }}\,\,{\bf{5}}{\bf{.25 \times 10}}{}^{{\bf{ - 7}}}}}{{\bf{2}}}\\{\bf{ = - 7}}{\bf{.83 \times 10}}{}^{{\bf{ - 7}}}{\bf{M/s}}\end{aligned}\)

Average rate of formation of \({{\bf{C}}_{\bf{8}}}{{\bf{H}}_{{\bf{12}}}}\) at 1600 s and the instantaneous rate of formation at 3200 s

Linked equation between average rate of dimerization and average rate of formation.

Equation:

\(\begin{aligned}{}{\bf{ = }}\frac{{{\bf{ - 1}}}}{{\bf{2}}}\frac{{{\bf{\Delta (}}{{\bf{C}}_{\bf{4}}}{{\bf{H}}_{\bf{6}}}{\bf{)}}}}{{{\bf{\Delta t}}}}\\{\bf{ = }}\frac{{{\bf{\Delta (}}{{\bf{C}}_{\bf{8}}}{{\bf{H}}_{{\bf{12}}}}{\bf{)}}}}{{{\bf{\Delta t}}}}\end{aligned}\)

Average rate of formation for \({{\bf{C}}_{\bf{8}}}{{\bf{H}}_{{\bf{12}}}}\) at 1600s and use rate of dimerization between 0s and 1600s

Equation:

\(\begin{aligned}{}{\bf{ = }}\frac{{{\bf{ - 1}}}}{{\bf{2}}}{\bf{ \times }}\frac{{{\bf{3}}{\bf{.10 \times 1}}{{\bf{0}}^{{\bf{ - 6}}}}{\bf{M/s}}}}{{}}\\{\bf{ = }}\frac{{{\bf{\Delta (}}{{\bf{C}}_{\bf{8}}}{{\bf{H}}_{{\bf{12}}}}{\bf{)}}}}{{{\bf{\Delta t}}}}\\{\bf{ = }}\frac{{{\bf{\Delta (}}{{\bf{C}}_{\bf{8}}}{{\bf{H}}_{{\bf{12}}}}{\bf{)}}}}{{{\bf{\Delta t}}}}\\{\bf{ = 1}}{\bf{.55 \times 1}}{{\bf{0}}^{{\bf{ - 6}}}}{\bf{M/s}}\end{aligned}\)

Linked equation between instantaneous rate of formation and instantaneous rate of dimerization.

Equation:

\(\begin{aligned}{}{\bf{ = }}\frac{{{\bf{ - 1}}}}{{\bf{2}}}\frac{{{\bf{d(}}{{\bf{C}}_{\bf{4}}}{{\bf{H}}_{\bf{6}}}{\bf{)}}}}{{{\bf{dt}}}}\\{\bf{ = }}\frac{{{\bf{d(}}{{\bf{C}}_{\bf{8}}}{{\bf{H}}_{{\bf{12}}}}{\bf{)}}}}{{{\bf{dt}}}}\end{aligned}\)

Instantaneous rate of formation for 3200s and use instantaneous rate of dimerization at 3200s into equation

\(\begin{aligned}{}{\bf{ = }}\frac{{{\bf{ - 1}}}}{{\bf{2}}}{\bf{ \times }}\frac{{{\bf{7}}{\bf{.83 \times 1}}{{\bf{0}}^{{\bf{ - 7}}}}{\bf{M/s}}}}{{\bf{1}}}\\{\bf{ = }}\frac{{{\bf{d(}}{{\bf{C}}_{\bf{8}}}{{\bf{H}}_{{\bf{12}}}}{\bf{)}}}}{{{\bf{dt}}}}\\{\bf{ = }}\frac{{{\bf{d(}}{{\bf{C}}_{\bf{8}}}{{\bf{H}}_{{\bf{12}}}}{\bf{)}}}}{{{\bf{dt}}}}\\{\bf{ = - 3}}{\bf{.92 \times 1}}{{\bf{0}}^{{\bf{ - 7}}}}{\bf{M/s}}\end{aligned}\)