Question

The method developed in Problem 14-11 was used for the routine determination of iron in 25.0-mL aliquots of groundwater. Express the concentration (as ppm Fe) in samples that yielded the accompanying absorbance data (1.00-cm cell). Calculate the relative standard deviation of the result. Repeat the calculation

assuming the absorbance data are means of three measurements.

(a) 0.143 (c) 0.068 (e) 1.512

(b) 0.675 (d) 1.009 (f) 0.546

Short answer

a) 0.170 ppm

b) 0.135 ppm

c) 0.178 ppm

d) 0.135 ppm

e) 0.170 ppm

f) 0.139 ppm

Step-by-step solution

Part(a) . Step 1: Given information

Absorbance data:-0.143

The standard deviation of the results obtained from the calibration curve = ${\mathrm{s}}_{\mathrm{c}}=\frac{{\mathrm{s}}_{\mathrm{r}}}{\mathrm{m}}\sqrt{\frac{1}{\mathrm{M}}+\frac{1}{\mathrm{N}}+\frac{{(\overline{{\mathrm{y}}_{\mathrm{c}}}-\overline{\mathrm{y}})}^2}{{\mathrm{m}}^2{\mathrm{S}}_{\mathrm{XX}}}}$

M = number of replicates

N = number of calibration points.

Part (a) Step 2: Calculation

$$\begin{gathered} \mathrm{y}=0.03949\mathrm{x}-0.00134 \\ 0.143=0.03949\mathrm{x}-0.00134 \\ 0.03949\mathrm{x}=0.14434 \\ \mathrm{x}=3.66\mathrm{ppm} \end{gathered}$$

$$\begin{gathered} {\mathrm{s}}_{\mathrm{c}}=\frac{0.0074}{0.03949}\sqrt{\frac{1}{1}+\frac{1}{6}+\frac{{(0.143-0.828)}^2}{{(0.03949)}^2\times 926}} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\sqrt{1+0.167+\frac{0.4692}{1.4441}} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\sqrt{1+0.167+0.325} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\times 1.221 \\ {\mathrm{s}}_{\mathrm{c}}=0.229\mathrm{ppm} \end{gathered}$$

If the experiment was replicated three times,

$$\begin{gathered} {\mathrm{s}}_{\mathrm{c}}=\frac{0.0074}{0.03949}\sqrt{\frac{1}{3}+\frac{1}{6}+\frac{{(0.143-0.828)}^2}{{(0.03949)}^2\times 926}} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\sqrt{0.333+0.167+\frac{0.4692}{1.4441}} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\sqrt{0.333+0.167+0.325} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\times 0.9083 \\ {\mathrm{s}}_{\mathrm{c}}=0.170\mathrm{ppm} \end{gathered}$$

Part (b) Step 1: Calculation

$$\begin{gathered} \mathrm{y}=0.03949\mathrm{x}-0.00134 \\ 0.675=0.03949\mathrm{x}-0.00134 \\ 0.03949\mathrm{x}=0.67634 \\ \mathrm{x}=17.13\mathrm{ppm} \end{gathered}$$

role="math" localid="1646374483543" $$\begin{gathered} {\mathrm{s}}_{\mathrm{c}}=\frac{0.0074}{0.03949}\sqrt{\frac{1}{1}+\frac{1}{6}+\frac{{(0.675-0.828)}^2}{{(0.03949)}^2\times 926}} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\sqrt{1+0.167+\frac{0.0234}{1.4441}} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\sqrt{1+0.167+0.016} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\times 1.087 \\ {\mathrm{s}}_{\mathrm{c}}=0.204\mathrm{ppm} \end{gathered}$$

If the experiment was replicated three times,

$$\begin{gathered} {\mathrm{s}}_{\mathrm{c}}=\frac{0.0074}{0.03949}\sqrt{\frac{1}{3}+\frac{1}{6}+\frac{{(0.675-0.828)}^2}{{(0.03949)}^2\times 926}} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\sqrt{0.333+0.167+\frac{0.0234}{1.4441}} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\sqrt{0.333+0.167+0.016} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\times 0.7183 \\ {\mathrm{s}}_{\mathrm{c}}=0.135\mathrm{ppm} \end{gathered}$$

Part (c) Step 1: Calculation

$$\begin{gathered} \mathrm{y}=0.03949\mathrm{x}-0.00134 \\ 0.068=0.03949\mathrm{x}-0.00134 \\ 0.03949\mathrm{x}=0.06934 \\ \mathrm{x}=1.76\mathrm{ppm} \end{gathered}$$

$$\begin{gathered} {\mathrm{s}}_{\mathrm{c}}=\frac{0.0074}{0.03949}\sqrt{\frac{1}{1}+\frac{1}{6}+\frac{{(0.068-0.828)}^2}{{(0.03949)}^2\times 926}} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\sqrt{1+0.167+\frac{0.5776}{1.4441}} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\sqrt{1+0.167+0.400} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\times 1.567 \\ {\mathrm{s}}_{\mathrm{c}}=0.294\mathrm{ppm} \end{gathered}$$

If the experiment was replicated three times,

$$\begin{gathered} {\mathrm{s}}_{\mathrm{c}}=\frac{0.0074}{0.03949}\sqrt{\frac{1}{3}+\frac{1}{6}+\frac{{(0.068-0.828)}^2}{{(0.03949)}^2\times 926}} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\sqrt{0.333+0.167+\frac{0.5776}{1.4441}} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\sqrt{0.333+0.167+0.400} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\times 0.949 \\ {\mathrm{s}}_{\mathrm{c}}=0.178\mathrm{ppm} \end{gathered}$$

Part (d) Step 1: Calculation

$$\begin{gathered} \mathrm{y}=0.03949\mathrm{x}-0.00134 \\ 1.009=0.03949\mathrm{x}-0.00134 \\ 0.03949\mathrm{x}=1.01034 \\ \mathrm{x}=25.58\mathrm{ppm} \end{gathered}$$

$$\begin{gathered} {\mathrm{s}}_{\mathrm{c}}=\frac{0.0074}{0.03949}\sqrt{\frac{1}{1}+\frac{1}{6}+\frac{{(1.009-0.828)}^2}{{(0.03949)}^2\times 926}} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\sqrt{1+0.167+\frac{0.0327}{1.4441}} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\sqrt{1+0.167+0.0227} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\times 1.091 \\ {\mathrm{s}}_{\mathrm{c}}=0.204\mathrm{ppm} \end{gathered}$$

If the experiment was replicated three times,

$$\begin{gathered} {\mathrm{s}}_{\mathrm{c}}=\frac{0.0074}{0.03949}\sqrt{\frac{1}{3}+\frac{1}{6}+\frac{{(1.009-0.828)}^2}{{(0.03949)}^2\times 926}} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\sqrt{0.333+0.167+\frac{0.0327}{1.4441}} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\sqrt{0.333+0.167+0.0227} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\times 0.7229 \\ {\mathrm{s}}_{\mathrm{c}}=0.135\mathrm{ppm} \end{gathered}$$

Part (e) Step 1: Calculation

$$\begin{gathered} \mathrm{y}=0.03949\mathrm{x}-0.00134 \\ 1.512=0.03949\mathrm{x}-0.00134 \\ 0.03949\mathrm{x}=1.51334 \\ \mathrm{x}=38.32\mathrm{ppm} \end{gathered}$$

$$\begin{gathered} {\mathrm{s}}_{\mathrm{c}}=\frac{0.0074}{0.03949}\sqrt{\frac{1}{1}+\frac{1}{6}+\frac{{(1.512-0.828)}^2}{{(0.03949)}^2\times 926}} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\sqrt{1+0.167+\frac{0.4678}{1.4441}} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\sqrt{1+0.167+0.324} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\times 1.221 \\ {\mathrm{s}}_{\mathrm{c}}=0.229\mathrm{ppm} \end{gathered}$$

If the experiment was replicated three times,

$$\begin{gathered} {\mathrm{s}}_{\mathrm{c}}=\frac{0.0074}{0.03949}\sqrt{\frac{1}{3}+\frac{1}{6}+\frac{{(1.512-0.828)}^2}{{(0.03949)}^2\times 926}} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\sqrt{0.333+0.167+\frac{0.4678}{1.4441}} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\sqrt{0.333+0.167+0.324} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\times 0.908 \\ {\mathrm{s}}_{\mathrm{c}}=0.170\mathrm{ppm} \end{gathered}$$

Part (f) Step 1: Calculation

$$\begin{gathered} \mathrm{y}=0.03949\mathrm{x}-0.00134 \\ 0.546=0.03949\mathrm{x}-0.00134 \\ 0.03949\mathrm{x}=0.54734 \\ \mathrm{x}=13.86\mathrm{ppm} \end{gathered}$$

$$\begin{gathered} {\mathrm{s}}_{\mathrm{c}}=\frac{0.0074}{0.03949}\sqrt{\frac{1}{1}+\frac{1}{6}+\frac{{(0.546-0.828)}^2}{{(0.03949)}^2\times 926}} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\sqrt{1+0.167+\frac{0.0795}{1.4441}} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\sqrt{1+0.167+0.0551} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\times 1.105 \\ {\mathrm{s}}_{\mathrm{c}}=0.207\mathrm{ppm} \end{gathered}$$

If the experiment was replicated three times,

$$\begin{gathered} {\mathrm{s}}_{\mathrm{c}}=\frac{0.0074}{0.03949}\sqrt{\frac{1}{3}+\frac{1}{6}+\frac{{(0.546-0.828)}^2}{{(0.03949)}^2\times 926}} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\sqrt{0.333+0.167+\frac{0.0795}{1.4441}} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\sqrt{0.333+0.167+0.0551} \\ {\mathrm{s}}_{\mathrm{c}}=0.1874\times 0.7451 \\ {\mathrm{s}}_{\mathrm{c}}=0.139\mathrm{ppm} \end{gathered}$$