Chapter 13, Problem 133RQ

On December 19, 2007, the T2 Laboratories, Inc., reactor exploded in a runaway reaction. The reaction of methyl cyclopentadienyl dimer and sodium produces sodium methyl cyclopentadiene and hydrogen:

The reactor has to be cooled when its temperature reaches $\text{182°C}$. On the day of the explosion, the cooling system malfunctioned and the reactor reached $\text{199°C}$, at which point a second, more exothermic reaction began to occur. The typical energy of activation for an organic reaction is about 50 kJ/mol. The heat of this reaction is 40 kJ mol^-1, and the heat capacity for an industrial scale reaction (i.e., 10,000 mol) can be $\text{7}\text{.5}\times {\text{10}}^{\text{6}}{\text{J °C}}^{\text{-1}}$

If the rare constant for a typical reaction is $\text{1}\times {\text{10}}^{\text{-5}}{\text{ s}}^{\text{-1}}\text{at 25°C}$, what will be the rate constant, if all the heat is supplied to the reaction all at once, without any external cooling? If the reaction were started at $\text{182°C}$, would the reaction reach $\text{199°C}$

if all the heat were released at once?