The conversion of cyclobutane (C4 H8) to ethylene (C2 H4)

Chapter 3, Problem 18SE

(choose chapter or problem)

Get Unlimited Answers
QUESTION:

The conversion of cyclobutane \(\left(\mathrm{C}_{4} \mathrm{H}_{8}\right)\) to ethylene \(\left(\mathrm{C}_{2} \mathrm{H}_{4}\right)\) is a first-order reaction. This means that the concentration of cyclobutane at time \(t\) is given by \(\ln\ C=\ln\ C_0-kt\), where \(C\) is the concentration at time \(t\), \(C_{0}\) is the initial concentration, \(t\) is the time since the reaction started, and \(k\) is the rate constant. Assume that \(C_0=0.2\mathrm{\ mol}/\mathrm{L}\) with negligible uncertainty. After 300 seconds at a constant temperature, the concentration is measured to be \(C=0.174\pm0.005\mathrm{\ mol}/\mathrm{L}\). 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 \(\mathrm{s}^{-1}\).

b. Find the relative uncertainty in \(k\).

c. The half-life \(t_{1 / 2}\) 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 \(t_{1/2}=(\ln\ 2)/k\). Using the result found in part (a), find the uncertainty in the half-life.

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

Equation Transcription:

Text Transcription:

 

C_4H_8

C_2H_4

t

ln C = ln C_0 - kt

C

C_0

 

k

C_0 = 0.2 mol/L

C = 0.174 pm 0.005 mol/L

s^-1

t_1/2

t_1/2 = (ln 2)/k

Questions & Answers

QUESTION:

The conversion of cyclobutane \(\left(\mathrm{C}_{4} \mathrm{H}_{8}\right)\) to ethylene \(\left(\mathrm{C}_{2} \mathrm{H}_{4}\right)\) is a first-order reaction. This means that the concentration of cyclobutane at time \(t\) is given by \(\ln\ C=\ln\ C_0-kt\), where \(C\) is the concentration at time \(t\), \(C_{0}\) is the initial concentration, \(t\) is the time since the reaction started, and \(k\) is the rate constant. Assume that \(C_0=0.2\mathrm{\ mol}/\mathrm{L}\) with negligible uncertainty. After 300 seconds at a constant temperature, the concentration is measured to be \(C=0.174\pm0.005\mathrm{\ mol}/\mathrm{L}\). 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 \(\mathrm{s}^{-1}\).

b. Find the relative uncertainty in \(k\).

c. The half-life \(t_{1 / 2}\) 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 \(t_{1/2}=(\ln\ 2)/k\). Using the result found in part (a), find the uncertainty in the half-life.

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

Equation Transcription:

Text Transcription:

 

C_4H_8

C_2H_4

t

ln C = ln C_0 - kt

C

C_0

 

k

C_0 = 0.2 mol/L

C = 0.174 pm 0.005 mol/L

s^-1

t_1/2

t_1/2 = (ln 2)/k

ANSWER:

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)

=

=

=

Add to cart


Study Tools You Might Need

Not The Solution You Need? Search for Your Answer Here:

×

Login

Login or Sign up for access to all of our study tools and educational content!

Forgot password?
Register Now

×

Register

Sign up for access to all content on our site!

Or login if you already have an account

×

Reset password

If you have an active account we’ll send you an e-mail for password recovery

Or login if you have your password back