The conversion of cyclobutane (C4 H8) to ethylene (C2 H4) is a first-order reaction. This means that the concentration of cyclobutane at time t is given by In C = In C0 − kt, where C is the concentration at time t, C0 is the initial concentration, t is the time since the reaction started, and k is the rate constant. Assume that C0 = 0.2 mol/L with negligible uncertainty. After 300 seconds at a constant temperature, the concentration is measured to be C = 0.174 ± 0.005 mol/L. Assume that time can be measured with negligible uncertainty.

a. Estimate the rate constant k, and find the uncertainty in the estimate. The units of k will be s-1.

b. Find the relative uncertainty in k.

c. The half-life t1/2 of the reaction is the time it takes for the concentration to be reduced to one-half its initial value. The half-life is related to the rate constant by t1/2= (In 2)/k. Using the result found in part (a), find the uncertainty in the half-life.

d. Find the relative uncertainty in the half-life.

Answer:

Step 1 of 4:

(a)

In this question, we are asked to estimate the rate constant and find the uncertainty in the estimate.

The concentration of cyclobutane at time t is given by where is the concentration at time is the initial concentration, is the time since the reaction is started, and is the rate constant.

Given concentration equation

………..(1)

Hence estimated rate constant is .

Uncertainty in the estimated rate constant k:

If is a measurement whose uncertainty is is small, and is a function of , then we can write

……………..(2)

Given

Given concentration equation

Substitute into the equation (2)

=

=

=

=

Substitute the value

= =

Hence uncertainty in

Therefore

Step of 2 of 4:

(b)

In this question, we are asked to find the relative uncertainty in

Relative uncertainty =

In our case

Relative uncertainty = ………………(3)

Substitute the values of and into the equation (3)

Relative uncertainty = = 0.2065

Relative uncertainty =

Hence relative uncertainty in =

We can therefore express the =