# The decomposition of nitrogen dioxide (NO2) into nitrogen ## Problem 19SE Chapter 3

Statistics for Engineers and Scientists | 4th Edition

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Problem 19SE

The decomposition of nitrogen dioxide (NO2) into nitrogen monoxide (NO) and oxygen is a second- order reaction. This means that the concentration C of NO2 at time t is given by 1/C = k t + 1/C0, where C0 is the initial concentration and k is the rate constant. Assume the initial concentration is known to be 0.03 mol/L exactly. Assume that time can be measured with negligible uncertainty.

a. After 40 s, the concentration C is measured to be 0.0023 ± 2.0 x 10-4 mol/L. Estimate the rate constant k, and find the uncertainty in the estimate.

b. After 50 s, the concentration C is measured to be 0.0018 ± 2.0 ×10-4 mol/L. Estimate the rate constant k, and find the uncertainty in the estimate.

c. Denote ihe estimates of the rate constant k in parts (a) and (b) by and respectively. The average is used as an estimate of k. Find the uncertainty in this estimate.

d. Find the value of c so that the weighted average has the smallest uncertainty.

Step-by-Step Solution:

Step 1 of 5:

Given, the decomposition of nitrogen dioxide ( into nitrogen monoxide (NO) and oxygen is a second order reaction.

This means that the concentration C of at time ‘t’ is given by: ….. (1)

Where, is the initial concentration and k is the rate constant.

Assume the initial concentration is known to be 0.03 mol/L exactly.

Step 2 of 5:

a). To estimate the rate constant k, and to find the uncertainty in the estimate.

After 40 s, the concentration C is measured to be The rate constant is defined by

From equation (1), we get ….(2)

Where, Substitute these values in the above equation. = 10.0362 L/s. mol

Therefore, the rate constant is k = 10.0362 L/s. mol.

Now, differentiate equation(2) with respect to C . (since = (t = 40)

= - 4725.8979

Therefore, = -4725.8979.

Now, the uncertainty is =  = 0.95 L/s. mol

Conclusion:

Therefore, the estimated value of rate constant is Here the uncertainty is 0.95.

Step 3 of 5:

b). To estimate the rate constant k, and to find the uncertainty in the estimate.

After 50 s, the concentration C is measured to be The rate constant is defined by

From equation (1), we get ….(2)

Where, Substitute these values in the above equation. = 10.4 L/s. mol

Therefore, the rate constant is k = 10.4 L/s. mol.

Now, differentiate equation(2) with respect to C . (since = (t = 50)

= = -6172.84

Therefore, = -6172.84.

Now, the uncertainty is =  = 1.2 L/s. mol

Conclusion:

Therefore, the estimated value of rate constant is Here the uncertainty is 1.2.

Step 4 of 5

Step 5 of 5

##### ISBN: 9780073401331

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