In a chemical reaction run at a certain temperature, the concentration C of a certain

Chapter 3, Problem 20

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In a chemical reaction run at a certain temperature, the concentration C of a certain reactant at time t is given by \(1 / C=k t+1 / C_{0}\), where \(C_{0}\) is the initial concentration and k is the rate constant. Assume the initial concentration is known to be \(0.04 \mathrm{~mol} / L\) exactly. Assume that time is measured with negligible uncertainty.

a. After \(30 \mathrm{~s}\), the concentration  is measured to be \(0.0038 \pm 2.0 \times 10^{-4} \mathrm{~mol} / \mathrm{L}\). Estimate the rate constant , and find the relative uncertainty in the estimate.

b. After \(50 \mathrm{~s}\), the concentration  is measured to be \(0.0024 \pm 2.0 \times 10^{-4} \mathrm{~mol} / \mathrm{L}\). Estimate the rate constant  and find the relative uncertainty in the estimate.

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

Equation Transcription:

     

   

     

   

   

   

Text Transcription:

1/C=kt+1/C0      

C0    

0.04 mol/L

30 s    

0.0038 \pm 2.0 x 10-4mol/L  

50 s  

0.0024 \pm 2.0 x 10-4mol/L

\widehat k1 and  \widehat k2  

\sqrt \widehat k1 \widehat  k2