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The decomposition of nitrogen dioxide (NO2) into nitrogen

Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi ISBN: 9780073401331 38

Solution for problem 19SE Chapter 3

Statistics for Engineers and Scientists | 4th Edition

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Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi

Statistics for Engineers and Scientists | 4th Edition

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1
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:

Answer :

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

Chapter 3, Problem 19SE is Solved
Step 5 of 5

Textbook: Statistics for Engineers and Scientists
Edition: 4
Author: William Navidi
ISBN: 9780073401331

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The decomposition of nitrogen dioxide (NO2) into nitrogen

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