The decomposition of nitrogen dioxide (NO2) into nitrogen monoxide (NO) and oxygen is a

Chapter 3, Problem 19

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The decomposition of nitrogen dioxide \(\left(\mathrm{NO}_{2}\right)\) into nitrogen monoxide \((\mathrm{NO})\) and oxygen is a second-order reaction. This means that the concentration \(\mathrm{C} \text { of } \mathrm{NO}_{2}\) at time  is given by \(1 / \mathrm{C}=k t+1 / \mathrm{Co}\) where \(\mathrm{Co}\) is the initial concentration and  is the rate constant. Assume the initial concentration is known to be \(0.03 \mathrm{~mol} / \mathrm{L}\) exactly. Assume that time can be measured with negligible uncertainty.

a. After \(40 \mathrm{~s}\), the concentration  is measured to be \(0.0023 \pm 2.0 \times 10^{-4} \mathrm{~mol} / \mathrm{L}\). Estimate the rate constant , and find the uncertainty in the estimate.

b. After \(50 \mathrm{~s}\), the concentration  is measured to be \(0.0018 \pm 2.0 \times 10^{-4} \mathrm{~mol} / \mathrm{L}\)Estimate the rate constant , and find the uncertainty in the estimate.

c. Denote the estimates of the rate constant  in parts (a) and (b) by \(\vec{k}_{1} \text { and } \widehat{k}_{2}\), respectively. The average \(\left(\vec{k}_{1}+\widehat{k}_{2}\right) 2\) is used as an estimate of . Find the uncertainty in this estimate.

d. Find the value of  so that the weighted average \(c \hat{k}_{1}+(1-c) \widehat{k}_{2}\) has the smallest uncertainty.

Equation Transcription:

   

   

   

   

     

   

     

 

   

Text Transcription:

(NO2)  

(NO)  

C of NO2    

1/C=kt +1/Co    

Co    

0.03 mol/L

40 s    

0.0023  \pm 2.0 x 10-4mol/L

50 s    

0.0018 \pm 2.0 x 10-4mol/L

\vec k1and \widehat k2  

(\vec k1 + \widehat k2)2    

C  \hat k1 + (1-c)\widehat k2