Question 3.WP.21: The isomerization of cis- and trans- 1-ethyl-2-methyl cyclop......
The isomerization of cis- and trans- 1-ethyl-2-methyl cyclopropane comes to equilibrium.
c i s\,{\overset{k_{1}}{\underset{k_{-1}}{\rightleftarrows}}}\,t r a n s
At 425.6 °C [t r a n s]_{\mathrm{eq}}/[c i s]_{\mathrm{eq}} = 2.79.
Throughout reaction the following data were obtained:
\operatorname{find}\,k_{1}{\mathrm{~and~}}k_{-1}.| \frac{[c i s]}{\mathrm{mol~dm}^{-3}} | 0.016 79 | 0.014 06 | 0.011 02 | 0.008 92 | 0.007 75 |
| \frac{time}{s} | 0 | 400 | 1000 | 1600 | 2100 |
This reaction is one where both the forward and back reactions are first order. Equation (3.91) can be used.
\log_{e} \left\{[A]_{actual}-[A]_{eq}\right\} =\log _{e}\left\{[A]_{0}-[A]_{eq}\right\} -(k_{1}+k_{-1})t (3.91)
At all times: [c i s]_{0}=[c i s]_{\mathrm{actual}}+[r a n s]_{\mathrm{actual}}=0.016~79~\mathrm{mol}~\mathrm{d}\mathrm{m}^{-3}
At equilibrium: [t r a n s]_{\mathrm{eq}}=2.79[c i s]_{\mathrm{eq}}
∴ 3.79\ [c i s]_{\mathrm{eq}}=0.016\,79\,\,\mathrm{mol~dm^{-3}} ∴ [c is]_{\mathrm{eq}}=4.43\times10^{-3}\ \mathrm{mol~dm^{-3}}
A plot of \log_{\mathrm{e}}\{[\mathrm{c}i s]_{t}-[c i s]_{\mathrm{cq}}\} versus time is linear with slope =-6.25\times10^{-4}\,s^{-1}
∴ k_{1}+k_{-1}=6.25\times10^{-4}\,{\bf s}^{-1}\quad K=k_{1}/k_{-1}=2.79
giving k_{1}=4.60\times10^{-4}~{\mathrm{s}}^{-1}~{\mathrm{and}}~k_{-1}=1.65\times10^{-4}~{\mathrm{s}}^{-1}
| \frac{[ci s]_{t}}{\mathrm{mol~dm}^{-3}} | 0.016 79 | 0.014 06 | 0.011 02 | 0.008 92 | 0.007 75 |
| \left\{{\frac{[c i s]_{t}-[c i s]_{e q}}{\mathrm{mol~dm}^{-3}}}\right\}. | 0.012 36 | 0.009 63 | 0.006 59 | 0.004 49 | 0.003 32 |
| \log_{\mathrm{e}}\left\{{\mathrm{[cis]}}_{t} – [cis]_{\mathrm{eq}}\right\} | -4.393 | -4.643 | -5.022 | -5.406 | -5.708 |
| \frac{time}{s} | 0 | 400 | 1000 | 1600 | 2100 |