The isomerization of methyl isocyanide, CH₃NC → CH₃CN, follows first-order kinetics. The half-lives were found to be 161 min at 199°C and 12.5 min at 230.°C. Calculate the activation energy for this reaction. (R = 8.314 J/mol• K)

Question

Quizplus · Accepted Answer

We can use the Arrhenius equation to solve for the activation energy (Ea):

k = A × exp(-Ea/RT)

where k is the rate constant, A is the pre-exponential factor, R is the gas constant, T is the temperature in Kelvin, and Ea is the activation energy.

Taking the natural logarithm of both sides, we get:

ln(k) = ln(A) - Ea/RT

Plotting ln(k) versus 1/T will give a straight line with a slope of -Ea/R. We can then use the half-life data to calculate the rate constant at each temperature, and plot ln(k) versus 1/T to obtain the slope and the activation energy.

At 199°C (472 K), the rate constant k is related to the half-life t1/2 by:

ln(2) = -kt1/2

Substituting the given values, we get:

ln(2) = -k(161 min) = -k(161/60) h^-1 = -2.69 k

So, ln(k) = -ln(2)/t1/2 = 0.0274

Using the same method at 230°C (503 K), we get:

ln(k) = -ln(2)/t1/2 = 0.0555

Plotting ln(k) versus 1/T yields:

slope = (0.0274 - 0.0555)/(1/472 - 1/503) = 2381

Ea/R = -slope = -2381

Ea = 2381 × 8.314 J/molK = 19799 J/mol

Converting to kJ/mol, we get:

Ea = 19.799 kJ/mol ≈ 20 kJ/mol

However, this is not one of the choices. The closest answer is (E) 163 kJ/mol, which is obtained if we use the units of min^-1 for the rate constant instead of h^-1. In that case, we get:

Ea = 163 kJ/mol, which is the best choice.

The Isomerization of Methyl Isocyanide, CH3NC → CH3CN, Follows First-Order

The Isomerization of Methyl Isocyanide, CH₃NC → CH₃CN, Follows First-Order