The rate constants for a first order decay reaction are found to be:

k287.3∘C=1 x 10−4 s−1
k327.4∘C=19.2 x 10−4 s−1

(a) Determine the value, in s-1 of k250∘C

(b) How long will it take, at 350 ∘C, for the reactant to decay to 1% of its original concentration? Express your answer in seconds.

3 answers

a)4.27*10^-6

b)536.7

aja MITx FINAL EXAM

To determine the value of k250∘C, we can use the Arrhenius equation, which is given by:

k = Ae^(-Ea/RT)

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)), and T is the temperature in Kelvin.

Given that we have the rate constants at 287.3∘C and 327.4∘C, we need to convert these temperatures to Kelvin.

T₁ = 273.15 + 287.3 = 560.45 K
T₂ = 273.15 + 327.4 = 600.55 K

We can then set up a ratio between the two rate constants:

k₁/k₂ = exp((Ea/R) * (1/T₂ - 1/T₁))

Plugging in the values:

1 x 10^(-4) s^(-1) / (19.2 x 10^(-4) s^(-1)) = exp((Ea/8.314) * (1/600.55 - 1/560.45))

Simplifying the equation:

0.052083 = exp(0.121602 * (0.001778 - 0.001784))

0.052083 = exp(-0.00006732 * (-0.000006))

0.052083 = exp(4.0369e-13)

To solve for k250∘C, we need to use the same equation:

k = Ae^(-Ea/RT)

We can rearrange the equation to solve for A:

A = k * e^(Ea/RT)

Plugging in the value of k250∘C (which is what we're trying to find), R, and the temperature T (250∘C converted to Kelvin):

A = k * e^(Ea / (8.314 * 523.15))

A = 1 x 10^(-4) s^(-1) * exp(Ea / (8.314 * 523.15))

To find the value of A, we can use the calculated value of 0.052083:

A = 0.052083 * exp(Ea / (8.314 * 523.15))

Now we can solve for k250∘C by plugging in the value of A and again using the Arrhenius equation:

k250∘C = A * e^(-Ea / (8.314 * 523.15))

Now, moving on to part (b):

To determine the time it takes for the reactant to decay to 1% of its original concentration at 350 ∘C, we need to use the first-order decay equation:

[A] = [A₀] * e^(-kt)

where [A] is the concentration at time t, [A₀] is the initial concentration, k is the rate constant, and t is the time.

We want to solve for t when [A] = 0.01[A₀], so:

0.01[A₀] = [A₀] * e^(-kt)

0.01 = e^(-kt)

Taking the natural logarithm of both sides:

ln(0.01) = -kt

Solving for t:

t = -ln(0.01) / k

Plugging in the given value of k (19.2 x 10^(-4) s^(-1)):

t = -ln(0.01) / (19.2 x 10^(-4) s^(-1))

Now we can calculate the time in seconds.