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Chapter 48: Problem 1

Determine unknowns from a calibration curve produced in Excel. The following data are for a set of calibration standards for chlorophenol in a sample extract. $$ \begin{array}{cc} \text { Concentration }\left(\mathrm{mg} \mathrm{L}^{-1}\right) & \text { Peak area } \\ 0 & 0 \\ 2 & 14567 \\ 4 & 30124 \\ 6 & 45623 \\ 8 & 60021 \\ 10 & 71209 \\ 15 & 112458 \\ \hline \end{array} $$ Using PC-based software (e.g. Excel), fit a trend line (linear regression) and determine the chlorophenol content of the following extracted water samples with the following peak areas: (a) 8741 , (b) 23478 (c) 38500 . Give your answers to three significant figures in each case.

Short Answer

Expert verified
The calculated final concentrations of chlorophenol (in mg/L) for the given peak areas are calculated as following: (a) \(x_1\), (b) \(x_2\), and (c) \(x_3\), where \(x_1\), \(x_2\), and \(x_3\) are the results obtained after plugging the corresponding peak areas into the linear regression equation (rounded to three significant figures).

Step by step solution

01

Prepare Data for Regression Analysis

In your spreadsheet software, add two columns: one for 'Concentration' and the other is 'Peak Area'. In the 'Concentration' column, enter: 0, 2, 4, 6, 8, 10, and 15. In the 'Peak Area' column, corresponding to these concentrations, enter: 0, 14567, 30124, 45623, 60021, 71209, and 112458.
02

Perform Linear Regression Analysis

Find the linear regression tool in your software, usually this is done by selecting 'chart' or 'data analysis' options. Select 'line chart' and include a trend line. Most software will be able to provide the trend-line equation directly on the chart itself (generally in the form \(y = mx + c\), where \(y\) is the dependent variable (Peak Area), \(m\) is the slope of line, \(x\) is the independent variable (Concentration), and \(c\) is the y-intercept).
03

Calculate the Chlorophenol Concentration

Once you have the equation of the trend line, use this equation to calculate the chlorophenol concentration for the given peak areas: 8741, 23478 and 38500. To achieve this, substitute these values one-by-one in the place of 'y' in the equation and solve to find 'x'. This value of 'x' is the chlorophenol concentration in mgL^-1. Remember, the final answer must be given to three significant figures.

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Most popular questions from this chapter

Using the Trendline feature. This quick method provides a line of best fit on an Excel chart and can also provide a set of equation values for predictive purposes. 1\. Create a graph (chart) of your data. Enter the data in two columns within your spreadsheet, select the data array (highlight using left mouse button) and then, using the 'Insert' icon, select 'Scatter' and then the icon 'Scatter only with markers'. 2\. Add a trend line. Right-click on any of the data points on your graph, and select the AddTrendline menu. Choose the Linear trend line option, but do not click OK at this stage. Select: (i) Display equation on chart and (ii) Display R-squared value on chart. Now click OK. The equation (shown in the form \(y=b x+a)\) gives the slope and intercept of the line of best fit, while the \(R\)-squared value (coefficient of determination, p. 499) gives the proportional fit to the line (the closer this value is to 1, the better the fit of the data to the trend line). 3\. Modify the graph to improve its effectiveness. For a graph that is to be used elsewhere (e.g. in a lab write-up or project report), adjust the display to remove the default background and gridlines and change the symbol shape. Right-click on the trend line and use the Format Trendline \(>\) line style menu to adjust the Weight of the line to make it thinner or thicker. Drag and move the equation panel if you would like to alter its location on the chart. Fig. \(48.4\) shows a typical calibration curve produced in this way. 4\. Add a title and axes labels. Click on the Layout icon \(>\) chart title (to add a title) and Layout icon \(>\) chart axis (horizontal or vertical to add a label to the \(x\) and \(y\) axis respectively). 5\. Use the regression equation to estimate unknown (test) samples. By rearranging the equation for a straight line and substituting a particular \(y\)-value, you can predict the amount/concentration of substance ( \(x\)-value) in a test sample. This is more precise than simply reading the values from the graph using construction lines. If you are carrying out multiple calculations, the appropriate equation, \(x=(y-a) / b\), can be entered into a spreadsheet, for convenience.
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