Question

A standard solution was put through appropriate dilutions to give the concentrations of iron shown next. The iron(II)-1,10, phenanthroline complex was then formed in 25.0-mL aliquots of these solutions, following which each was diluted to 50.0 mL. The following absorbances (1.00-cm cells) were recorded at 510 nm:

$$\begin{gathered} \underline{\mathrm{Fe}\left(\mathrm{II}\right)\mathrm{Concentration}\mathrm{in} \\ \mathrm{Original}\mathrm{Solutions},\mathrm{ppm}{\mathrm{A}}_{510}} \\ 4.000.160 \\ 10.00.390 \\ 16.00.630 \\ 24.00.950 \\ 32.01.260 \\ 40.01.580 \end{gathered}$$

(a) Plot a calibration curve from these data.

(b) Use the method of least squares to find an equation relating absorbance and the concentration

of iron(II).

(c) Calculate the standard deviation of the slope and intercept.

Short answer

(a)

(b) $y=0.03949x-0.00134$

(c) Standard deviation of the slope and intercept are $0.00024$and $0.0059$.

Step-by-step solution

Part (a) Step 1: Given information

The following absorbances (1.00-cm cells) were recorded at 510 nm:

$$\begin{gathered} \underline{\mathrm{Fe}\left(\mathrm{II}\right)\mathrm{Concentration}\mathrm{in} \\ \mathrm{Original}\mathrm{Solutions},\mathrm{ppm}{\mathrm{A}}_{510}} \\ 4.000.160 \\ 10.00.390 \\ 16.00.630 \\ 24.00.950 \\ 32.01.260 \\ 40.01.580 \end{gathered}$$

Part (a) Step 2: Calibration curve 

A calibration curve or standard curve is a method used in analytical chemistry to determine the concentration of unknown sample by comparing to a set of samples with known concentrations.

Part (a) Step 3: Plot for absorbance at 510 nm versus Fe(II) concentration 

Absorbance is the dependent variable and Fe(II) concentration is the independent variable.

The plot for absorbance at 510 nm versus Fe(II) concentration in the original solution is:

Part (b) Step 1: Method of least squares

For equations:

$$\begin{gathered} S_{yy}=\sum _{}^{}{\left(y_i-\overline{y}\right)}^2 \\ {S}_{xx}=\sum _{}^{}{\left(x_i-\overline{x}\right)}^2 \\ {S}_{xy}=\sum _{}^{}\left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right) \end{gathered}$$

The slope of the line,$m=\frac{S_{xy}}{S_{xx}}$

The intercept,$b=\overline{y}-m\overline{x}$

Part (b) Step 2: Relationship between Fe(II) concentration and absorbance  

$$\begin{gathered} S_{yy}=\sum _{}^{}{\left(y_i-\overline{y}\right)}^2=1.44428 \\ {S}_{xx}=\sum _{}^{}{\left(x_i-\overline{x}\right)}^2=926 \\ {S}_{xy}=\sum _{}^{}\left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)=36.57 \end{gathered}$$

$$\begin{gathered} m=\frac{S_{xy}}{S_{xx}}=\frac{36.57}{923}=0.03949 \\ b=0.828-0.03949\times 21 \\ =-1.34\times {10}^{-3} \end{gathered}$$

Therefore, the equation for the relationship between Fe(II) concentration and absorbance is:

localid="1650369875586" $y=0.03949x-0.00134$

 Part (c) Step 1: Standard deviation  

Standard deviation about regression:

$S_r=\sqrt{\frac{S_{yy}-m^2{S}_{xx}}{N-2}}$ ... (1)

The standard deviation of the slope:

$S_m=\sqrt{\frac{{S_r}^2}{S_{xx}}}$ ... (2)

The standard deviation of the intercept:

$S_b=\sqrt[S_r]{\frac{\sum _{}^{}{x_i}^2}{N\sum _{}^{}{x_i}^2-{\left(\sum _{}^{}{x}_i\right)}^2}}$ ... (3)

Part (c) Step 2: Calculate standard deviation of the slope and intercept 

	x	xi2
	4.00	16
	10.00	100
	16.00	256
	24.00	576
	32.00	1024
	40.00	1600
Sum	126	3572

From equation (1), (2) and (3):

$\begin{array}{rcl}{S}_r& =& \sqrt{\frac{1.14428-{\left(0.03949\right)}^2\times 926}{6-2}}\\ & =& 0.0074\\ S_m& =& \sqrt{\frac{{\left(0.0074\right)}^2}{926}}\\ & =& 0.00024\end{array}$

$\begin{array}{rcl}{S}_b& =& \sqrt[0.0074]{\frac{3572}{6\times 3572-{\left(126\right)}^2}}\\ & =& 0.0074\times 0.8018\\ & =& 0.0059\end{array}$