Chapter 7, Problem 1SE

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 L = 1 cm. Measurements of A are made at various concentrations. The data are presented in the following table.

1. a. Let $A={\hat{\beta }}_0+{\hat{\beta }}_{1}C$ be the equation of the least-squares line for predicting absorbance (A) from concentration (C). Compute the values of ${\hat{\beta }}_0$ and ${\hat{\beta }}_1$.
2. b. What value does the Beer–Lambert law assign to β₀?
3. c. What physical quantity does ${\hat{\beta }}_1$ estimate?
4. d. Test the hypothesis H₀ : β₀ = 0. Is the result consistent with the Beer–Lambert law?

To determine

Find the values of ${\widehat{\beta }}_0$ and ${\widehat{\beta }}_1$.

Answer to Problem 1SE

The values of ${\widehat{\beta }}_0=-\text{0}\text{.0390}$ and ${\widehat{\beta }}_1=\text{1}\text{.0169}$.

Explanation of Solution

Calculation:

The given information is that 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.

Least-squares line:

Software Procedure:

Step-by-step procedure to obtain the least-squares line using the MINITAB software is given below:

• Choose Stat > Regression > Regression > Fit Regression Model.
• In Responses, enter “Absorbance”.
• In Continuous predictors, enter “Concentration”.
• Check Results.
• In Display of results, choose Simple tables.
• Click OK.

Output using the MINITAB software is given below:

From the MINITAB output, ${\widehat{\beta }}_0=-\text{0}\text{.0390}$ and ${\widehat{\beta }}_1=\text{1}\text{.0169}$.

To determine

Find the value does the Beer-Lambert law assign to ${\beta }_0$.

Explanation of Solution

Calculation:

Here, the variable C is independent, the variable A is dependent, the length L is constant and M is the molar absorption coefficient.

The equation of the Beer-Lambert law is,

$A=MLC$ (1)

The equation of the least-squares line for predicting absorbance (A) from concentration (C) is,

$A={\widehat{\beta }}_0+{\widehat{\beta }}_{1}C$ (2)

On comparing equation (1) and (2),

$\begin{array}{c}MLC={\widehat{\beta }}_0+{\widehat{\beta }}_{1}C\\ MC={\widehat{\beta }}_0+{\widehat{\beta }}_{1}C\left(\text{Since, }L=1\right)\end{array}$

Thus, the coefficient ${\widehat{\beta }}_1$ estimates the molar absorption coefficient M. Therefore, the Beer-Lambert law assign to ${\beta }_0$ is 0.

To determine

Find the physical quantity does ${\widehat{\beta }}_1$ estimate.

Explanation of Solution

Justification:

From part b., comparing equation (1) and (2) is, $MC={\widehat{\beta }}_0+{\widehat{\beta }}_{1}C$.

Therefore, the coefficient ${\widehat{\beta }}_1$ estimates the molar absorption coefficient M.

To determine

Test the hypothesis $H_0:{\beta }_0=0$.

Check whether the result is consistent with the Beer-Lambert law.

Answer to Problem 1SE

Yes, the result is consistent with the Beer-Lambert law.

Explanation of Solution

Calculation:

State the null and alternative hypotheses.

Null hypothesis:

$H_0:{\beta }_0=0$

Alternative hypothesis:

$H_1:{\beta }_0\ne 0$

From the MINITAB obtained in part a., the test statistic for slope is –0.43 and the P- value is 0.697.

Conclusion:

Here, the P-value is not small.

That is, the P-value is greater than the level of significance, 0.05.

Therefore, the null hypothesis is not rejected.

Hence, it can be concluded that ${\beta }_0=0$.

Thus, the result is consistent with the Beer-Lambert law.