Let's analyze the problem step by step.
The rate constant for the reaction depends on temperature according to the Arrhenius equation:
k=Aexp(−),where
Ea is the activation energy,
R is the universal gas constant,
T is the absolute temperature, and
A is the pre-exponential factor.
Assuming that the cooking (boiling) time is inversely proportional to the rate constant (i.e., a faster reaction means a shorter time to cook), we can write:
t∝ We're given:
At sea level (temperature
T1 ), the time is 4.0 minutes.
At the mountain top (temperature
T2 ), the time is 8.0 minutes.
Therefore, the ratio of the cooking times is:
==2 Since time is inversely proportional to the rate constant, we have:
==2.Now, express the rate constants using the Arrhenius equation:
Equate this to 2 :
2=exp[−(−)]Take the natural logarithm on both sides:
ln2=−(−)Rearrange to solve for the activation energy
Ea :
Ea=−Rln2(−)−1 Notice that
−=.So, substituting gives:
Since the sea level temperature
T1 is higher than the mountain top boiling temperature
T2, the term
T2−T1 is negative. The negative sign in front will cancel that, resulting in a positive activation energy:
Ea=Rln2⋅.Using
ln2≈0.693, we have:
Ea=0.693R.Comparing this with the given options, we see that it matches:
Option D:
0.693R.
So, the activation energy for the reaction is given by Option D.