P35.10 (Challenging) The first-order thermal decomposition of
chlorocyclohexane is as follows: \(\mathrm{C}_{6} \mathrm{H}_{11}
\mathrm{Cl}(g) \rightarrow\) \(\mathrm{C}_{6} \mathrm{H}_{10}(g)+\mathrm{HCl}(g)
.\) For a constant volume system the following total pressures were measured as
a function of time:
$$\begin{array}{rccr}\text { Time }(s) & P(\text { Torr }) & \text { Time }(s)
& P(\text { Torr }) \\\\\hline 3 & 237.2 & 24 & 332.1 \\\6 & 255.3 & 27 &
341.1 \\\9 & 271.3 & 30 & 349.3 \\\12 & 285.8 & 33 & 356.9 \\\15 & 299.0 & 36
& 363.7 \\\18 & 311.2 & 39 & 369.9 \\\21 & 322.2 & 42 & 375.5\end{array}$$
a. Derive the following relationship for a first-order reaction:
\\[P\left(t_{2}\right)-P\left(t_{1}\right)=\left(P\left(t_{\infty}\right)-P\left(t_{0}\right)\right)
e^{-k t_{1}}\left(1e^{-k\left(t_{2}-t_{1}\right)}\right)\\] In this relation,
\(P\left(t_{1}\right)\) and \(P\left(t_{2}\right)\) are the pressures at two
specific times; \(\mathrm{P}\left(t_{0}\right)\) is the initial pressure when
the reaction is initiated, \(P\left(t_{\infty}\right)\) is the pressure at the
completion of the reaction, and \(k\) is the rate constant for the reaction. To
derive this relationship do the following:
i. Given the first-order dependence of the reaction, write the expression for
the pressure of chlorocyclohexane at a specific time \(t_{1}\)
ii. Write the expression for the pressure at another time
\(t_{2},\) which is equal to \(t_{1}+\Delta\) where delta is a fixed quantity of
time.
iii. Write expressions for \(P\left(t_{\infty}\right)-P\left(t_{1}\right)\) and
\(P\left(t_{\infty}\right) P\left(t_{2}\right)\)
iv. Subtract the two expressions from part (iii).
b. Using the natural log of the relationship from part (a) and the data
provided in the table given earlier in this problem, determine the rate
constant for the decomposition of chlorocyclohexane. (Hint: Transform the data
in the table by defining \(t_{2}-t_{1}\) to be a constant value, for example, 9
s.
