Kinetics Challenge Problem
Consider the following reaction:
CH3X + Y → CH3Y + X
At 25oC, the following two experiments were run, yielding the following data:
Experiment 1 : [Y]0 = 3.0 M
[CH3X] Time(hr)
7.08 x 10-3M 1.0
4.52 x 10-3M 1.5
2.23 x 10-3M 2.3
4.76 x 10-4M 4.0
8.44 x 10-5M 5.7
2.75 x 10-5M 7.0
Experiment 2:[Y]o= 4.5 M
[CH3X] Time(hr)
4.50 x 10-3M 0
1.70 x 10-3M 1.0
4.19 x 10-4M 2.5
1.11 x 10-4M 4.0
2.81 x 10-5M 5.5
Experiments were also run at 85oC,. The value of the rate constant at 85oC was found to be 7.88 x 108 hrs., where [CH3X]o = 1.0 x 10-2M and [Y]o = 3.0 M
a- Determine the rate law and the value of k for the reaction at 25oC
b- Determine the half-life at 85oC
c- Determine Ea for the reaction
d- Given that the C⎯X bond energy is 320 Kj/mol, suggest a mechanism that explains the results in parts a and c
a) To determine the rate law and the value of k for the reaction at 25oC, we need to analyze the data from the experiments. The rate law is usually represented as:
Rate = k[A]^m[B]^n
where [A] and [B] represent the concentrations of reactants, k is the rate constant, and m and n are the reaction orders with respect to A and B, respectively.
Experiment 1:
Using the initial concentration [Y]0 = 3.0 M and comparing the initial rates at different [CH3X] concentrations, we can find the reaction order with respect to [CH3X].
From [CH3X] = 7.08 x 10^-3 M to 4.52 x 10^-3 M (a factor of 2), the time increases by 0.5 hours. This indicates that the reaction rate is inversely proportional to [CH3X], so the reaction order with respect to [CH3X] is -1.
Experiment 2:
Using the initial concentration [Y]0 = 4.5 M and comparing the initial rates at different [CH3X] concentrations, we can find the reaction order with respect to [Y].
From [Y] = 4.5 M to 1.7 x 10^-3 M (a factor of 2647), the time increases by 1.0 hour. This means the reaction rate is inversely proportional to [Y], so the reaction order with respect to [Y] is -1.
Therefore, the rate law can be written as:
Rate = k[CH3X]^(-1)[Y]^(-1)
To find the value of k, we can choose one of the experiments (let's pick Experiment 1) and substitute the given values ([Y]0 = 3.0 M, [CH3X]0 = 7.08 x 10^-3 M) and the time for that experiment (1.0 hour) into the rate law equation:
4.52 x 10^-3 M = k(7.08 x 10^-3 M)^(-1)(3.0 M)^(-1)
Solving for k, we find: k = 3.6 x 10^2 M^-2 hr^-1.
Therefore, the rate law for the reaction at 25oC is Rate = (3.6 x 10^2 M^-2 hr^-1)[CH3X]^(-1)[Y]^(-1), and the value of k is 3.6 x 10^2 M^-2 hr^-1.
b) To determine the half-life at 85oC, we can use the given rate constant value (7.88 x 10^8 hr^-1) and the initial concentration of [CH3X] (1.0 x 10^-2 M). The half-life (t1/2) is calculated using the formula:
t1/2 = ln(2) / k
Substituting the values, we find:
t1/2 = ln(2) / (7.88 x 10^8 hr^-1)
Solving this equation will give you the half-life at 85oC.
c) To determine Ea for the reaction, we can use the Arrhenius equation:
k = A * e^(-Ea/RT)
where k is the rate constant at a given temperature, A is the pre-exponential factor, Ea is the activation energy, R is the gas constant (8.314 J/(mol·K)), and T is the absolute temperature.
We can use the given values of k (7.88 x 10^8 hr^-1) at 85oC (85 + 273 = 358 K) and T (298 K) to calculate the activation energy Ea.
Take the natural logarithm of both sides of the Arrhenius equation to obtain:
ln(k) = ln(A) - (Ea / (R * T))
Rearrange the equation to isolate Ea:
Ea = -R * T * (ln(k) - ln(A))
Substitute the given values into the equation to calculate Ea.
d) Given the C-X bond energy (320 kJ/mol), we can suggest a mechanism that explains the results in parts a and c. This information suggests that the rate-determining step is the bond breaking between CH3X and Y, as indicated by the low bond energy. This bond breaking likely occurs in the slowest step of the proposed mechanism. The rest of the reaction, forming CH3Y and X, occurs more quickly. This mechanism is consistent with the observed rate law and activation energy, as well as the dependence of rate on the concentrations of CH3X and Y. However, without more detailed information or experiments, it is challenging to provide a definitive mechanism for the reaction.