Statistics For Engineers And Scientists - 4 Edition - Chapter 3.4 - Problem 9e
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# 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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