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

Solution for problem 9E Chapter 3.4

The Beer-Lambert law relates the absorbance A of a

Statistics for Engineers and Scientists | 4th Edition


Problem 9E

The Beer-Lambert law relates the absorbance \(A\) of a solution to the concentration \(C\) of a species in solution by A = MLC, where \(L\) is the path length and \(M\) is the molar absorption coefficient. Assume that \(C=1.25\pm0.03\mathrm{\ mol}/\mathrm{cm}^3\), \(L=1.2\pm0.1\mathrm{\ cm}\), and \(A=1.30 \pm 0.05\).

a. Estimate \(M\) and find the uncertainty in the estimate.

b. Which would provide a greater reduction in the uncertainty in \(M\): reducing the uncertainty in \(C\) to \(0.01\mathrm{\ mol}/\mathrm{cm}^3\), reducing the uncertainty in \(L\) to 0.05 cm, or reducing the uncertainty in \(A\) to 0.01?

Equation Transcription:

Text Transcription:

 

A

C

L

M

C = 1.52 pm 0.03 mol/cm^3

L = 1.2 pm 0.1 cm

A = 1.30 pm 0.05

0.01 mol/cm^3

Accepted Solution
Step-by-Step Solution:
Step 1 of 3

Solution 9E

Step1 of 3:

We have Beer-Lambert law in that A be the absorbance of  a solution to the concentration C of a species and Let A = MLC.

Where,

L = Path length.

   = 1.2 ± 0.1 cm.

M = the molar absorption coefficient.

C = 1.25 ± 0.03 mol/cm3.

A = 1.30 ± 0.05.

Here our goal is:

a).We need to estimate M and find the uncertainty in the estimate.

b).We need to check Which would provide a greater reduction in the uncertainty in M: reducing the uncertainty in C to 0.01 mol/cm3, reducing the uncertainty in L to 0.05 cm, or reducing the uncertainty in A to 0.01?


Step2 of 3:

a).

We have C = 1.25, L = 1.2, A = 1.30 and

Let

   A = MLC

M =

             =

      =

       = 0.8667

Hence, M = 0.8667.

1).Consider,

M =

Differentiate above equation with respect to “A” then

   

             =

       =(1)

          =

  =

     = 0.6667

Hence, = 0.6667.

 

2).Consider,

M =

 Differentiate above equation with respect to “L” then

   

             =

               =

                  =

             =()

 

         =

                  =

        =

       = -0.7222

Hence, = -0.7222.

3).Consider,

M =

 Differentiate above equation with respect to “C” then

   

             =

               =

                  =

             =()

        =

                  =

        =

       = -0.6933

Hence, = -0.6933.

The estimate of uncertainty is given by

               

                                                         

               

                    =

                    =

                    = 0.0777

Hence, = 0.0777.

Therefore, The estimate of uncertainty M = 0.86670.0777.


Step3 of 3:

b).

From part (a), we have

 

Where,

1).=

         = 0.6667

2).=

             = -0.7222

3).=

             =  -0.6933

Here,

If then

If then

If then

As reducing the uncertainty in L to 0.05 cm provides the greatest reduction.


Chapter 3.4, Problem 9E is Solved

Step 2 of 3


Step 3 of 3

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The Beer-Lambert law relates the absorbance A of a