To solve this problem, we can make use of the Arrhenius equation, which relates the rate constant of a reaction to its temperature. The Arrhenius equation is given by:
k=A‌exp(−‌)where:
k is the rate constant
A is the pre-exponential factor (frequency factor)
Ea is the activation energy
R is the gas constant
T is the temperature in Kelvin
For two different temperatures, the Arrhenius equation can be written as:
‌=exp(‌(‌−‌))Given:
‌k1=0.004 s−1‌ at ‌T1=500K‌k2=0.014 s−1‌Ea=18.231‌kJ∕mol‌ (which is ‌18,231J∕mol‌ ) ‌‌R=8.314J∕mol‌K We need to find the new temperature
T2. Rearranging the Arrhenius equation to solve for
T2, we get:
ln(‌)=‌(‌−‌)Substituting the given values:
ln(‌)=‌(‌−‌)Calculating the left-hand side:
ln(‌)=ln(3.5)≈1.2528Thus, the equation becomes:
1.2528=‌(‌−‌)Simplifying further:
‌1.2528=‌×(‌−‌)‌1.2528=2,192.6×(‌−‌)‌‌−‌=‌≈0.0005714‌‌=‌−0.0005714‌‌≈0.002−0.0005714=0.0014286‌T2=‌≈700KThus, the correct option is Option C: 700 K .