The activation energy is the minimum energy required for a reaction to proceed. In other words, it is the energy that must be overcome in order for the chemical reaction to happen. It can be derived in many ways and one of which is the use of the reaction rate constants and changes in temperature. The equation can be written as:

`ln [(k2)/(k1)] = (Ea)/(R) ((1)/(T1) - (1)/(T2))`

Where:

k1 and k2 are reaction rate constants

R = gas constant 8.314 J/mol-K

T1 and T2 are absolute temperature (K)

Ea = energy of activation = 33kJ/mol = 33000j/mol

The problem is stating that the reaction rate increase by a factor of 2. This means that if k1 is 1 then k2 would be 1x2 or 2. Therefore k2/k1 = 2.

Now we can solve the problem since it is asking for the change in temperature when the k1 is increased by a factor of 2.

We try to transform the equation first and then substitute the given values.

`ln [(k2)/(k1)] = (Ea)/(R) ((1)/(T1) - (1)/(T2))`

`ln [(2)/(1)] = (33000)/(8.314) ((1)/(300) - (1)/(T2))`

`0.6931 = 3969.21 ((1)/(300) - (1)/(T2))`

` ` `(0.6931)/(3969.21) =((1)/(300) - (1)/(T2))`

`0.0001746 =((1)/(300) - (1)/(T2))`

`(1)/(T2) = (1)/(300) - 0.0001746`

`(1)/(T2) = 0.0033333 - 0.0001746`

`(1)/(T2) = 0.0031587`

`T2 = (1)/(0.0031587)`

**T2 = 316.59 K**

:)

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