A reaction rate increases by a factor of 815 in the presence of a catalyst at 37 degrees C. The activation energy of the original pathway is 106 kJ/mol. What is the activation energy of the new pathway, all other factors being
equal?
To determine the activation energy of the new pathway, we can use the Arrhenius equation, which relates the rate constant (k) of a reaction to the temperature (T) and activation energy (Ea).
The Arrhenius equation is given by:
k = Ae^(-Ea/RT)
Where:
- k is the rate constant
- A is the pre-exponential factor (a constant that depends on the particular reaction)
- Ea is the activation energy
- R is the gas constant (8.314 J/(K*mol))
- T is the temperature in Kelvin
We are given that the rate increases by a factor of 815 in the presence of a catalyst at 37 degrees C. The factor by which the rate increases is directly proportional to the rate constant. Therefore, we can write:
k_new = 815 * k_original
Let's solve for the activation energy of the new pathway, Ea(new):
k_new = A_new * e^(-Ea(new)/RT)
And k_original:
k_original = A_original * e^(-Ea(original)/RT)
Since the other factors remain equal, the pre-exponential factor (A) and the temperature (T) will cancel out when dividing the two equations. We can rewrite the equation as:
k_new / k_original = e^((Ea(original) - Ea(new))/RT) [taking the natural logarithm of both sides]
ln(k_new / k_original) = (Ea(original) - Ea(new))/RT
Finally, solving for Ea(new), we have:
Ea(new) = Ea(original) - RT * ln(k_new / k_original)
Now, we can plug in the known values:
Ea(original) = 106 kJ/mol (given)
R = 8.314 J/(K*mol) (gas constant)
T = 37 degrees C + 273.15 = 310.15 K (convert temperature to Kelvin)
k_new / k_original = 815 (given)
Ea(new) = 106 kJ/mol - 8.314 J/(K*mol) * 310.15 K * ln(815)
Calculating this expression will give us the activation energy of the new pathway.
The reaction rate increases by a factor of 815 in the presence of a catalyst. This means that the rate constant for the catalyzed pathway is 815 times greater than the rate constant for the uncatalyzed pathway.
To determine the activation energy of the new pathway, we can use the Arrhenius equation:
k = A * e^(-Ea / (R * T))
where:
- k is the rate constant
- A is the pre-exponential factor
- Ea is the activation energy
- R is the gas constant (8.314 J/(mol K))
- T is the temperature in Kelvin
Since all other factors are equal, we can assume that the pre-exponential factor and the temperature remain the same for both pathways (uncatalyzed and catalyzed).
Let's denote the activation energy of the catalyzed pathway as Ea_new.
Since the rate constant for the catalyzed pathway is 815 times greater than the rate constant for the uncatalyzed pathway, we can write:
k_new = 815 * k_uncatalyzed
Plugging these values into the Arrhenius equation, we get:
A * e^(-Ea_new / (R * T)) = 815 * (A * e^(-Ea_uncatalyzed / (R * T)))
The pre-exponential factor (A) and the temperature (T) are the same for both pathways, so we can cancel them out:
e^(-Ea_new / (R * T)) = 815 * e^(-Ea_uncatalyzed / (R * T))
Taking the natural logarithm of both sides, we get:
-Ea_new / (R * T) = ln(815 * e^(-Ea_uncatalyzed / (R * T)))
Simplifying:
Ea_new = -ln(815 * e^(-Ea_uncatalyzed / (R * T))) * (R * T)
Substituting the values:
- Ea_uncatalyzed = 106 kJ/mol
- R = 8.314 J/(mol K)
- T = 37 + 273 = 310 K
Ea_new = -ln(815 * e^(-106000 / (8.314 * 310))) * (8.314 * 310)
Calculating the expression inside the logarithm:
Ea_new = -ln(815 * e^(-40.648)) * 2572.84
Finally,
Ea_new = -ln(815 * 0.0000000000000000000000000000000000000000000000000000000000000000000000000008749) * 2572.84
Calculating ln(815 * 0.0000000000000000000000000000000000000000000000000000000000000000000000000008749):
Ea_new ≈ -68.716 * 2572.84
Ea_new ≈ -176,672.928 kJ/mol
Therefore, the activation energy of the new pathway, all other factors being equal, is approximately -176,672.928 kJ/mol.