Final answer:
To find the temperature at which a reaction goes twice as fast, we can use the Arrhenius equation. The temperature can be determined by rearranging the equation and solving for the unknown temperature. The activation energy and rate constant at the initial temperature are used in the calculation.
Step-by-step explanation:
The activation energy of a reaction determines the temperature at which the reaction occurs. To find the temperature at which the reaction would go twice as fast, we can use the Arrhenius equation. According to the Arrhenius equation, the rate constant (k) is proportional to the exponential of the negative activation energy (Ea) divided by the product of the Gas constant (R) and the temperature in Kelvin (T). Mathematically, it can be written as:
k = A * exp(-Ea / (R * T))
For the reaction to go twice as fast, we need to find the temperature (T2) at which the rate constant (k2) is double the rate constant (k1) at the initial temperature (T1). Therefore, we can set up the following equation:
k2 = 2 * k1
A * exp(-Ea / (R * T2)) = 2 * A * exp(-Ea / (R * T1))
By canceling out the frequency factor (A) and simplifying the equation, we get:
exp(-Ea / (R * T2)) = 2 * exp(-Ea / (R * T1))
Taking the natural logarithm of both sides of the equation, we get:
-Ea / (R * T2) = ln(2) -Ea / (R * T1)
By rearranging the equation and solving for T2, we can find the temperature at which the reaction would go twice as fast.