c
\(\log \,\frac{{{k_2}}}{{{k_1}}}\, = \,\frac{{{E_a}}}{{2.303\,R}}\left[ {\frac{{{T_2} - {T_1}}}{{{T_1}{T_2}}}} \right]\)

or \(2.303\,\log \,\frac{{{k_2}}}{{{k_1}}}\, = \,\frac{{{E_a}}}{R}\,\left[ {\frac{{{T_2} - {T_1}}}{{{T_1}{T_2}}}} \right]\)

\(2.303\,\log \,\left[ {\frac{{1.667\, \times \,{{10}^{ - 4}}}}{{1.667 \times \,{{10}^{ - 6}}}}} \right]\, = \) \(\, - \frac{{{E_a}}}{R}\left[ {\frac{1}{{1844}} - \frac{1}{{1000}}} \right]\)

\(2.303\, \times \,2\, = \,\frac{{{E_a}}}{R} \times \,\frac{{844}}{{1844 \times 1000}}\)   .... \((1)\)

\(\therefore \,\frac{{{E_a}}}{R}\, = \,\frac{{4.606\, \times \,1844 \times 1000}}{{844}}\)

\(2.303\,\log \,\left[ {\frac{{{k_3}}}{{1.667\, \times \,{{10}^{ - 6}}}}} \right]\,\) \( = \,\frac{{{E_a}}}{R} \times \,\frac{{1423 - 1000}}{{1423 \times 1000}}\)

\( = \,\frac{{{E_a}}}{R}\, \times \,\frac{{423}}{{1423 \times 1000}}\)        .... \((2)\)

Dividing equation \((2)\) by equation \((1)\)

\(\frac{{\log \left[ {\frac{{{k_3}}}{{1.667 \times {{10}^{ - 6}}}}} \right]}}{2}\)

\( = \,\frac{{423}}{{1423 \times 1000}}\, \times \,\frac{{1844 \times 1000}}{{844}}\)

\(\therefore \,\,\log \left[ {\frac{{{k_3}}}{{1.667 \times {{10}^{ - 6}}}}} \right]\)

\( = \,2\, \times \,\frac{{423 \times 1844}}{{1423 \times 844}}\, = \,1.299\)

On taking antilog , \(k_3=19.9\)

\({k_3}\, = \,19.9\, \times 1.667 \times \,{10^{ - 6}}\, = \,3.318\, \times \,{10^{ - 5}}\,{s^{ - 1}}\)