1

3. Activity: Data Collection in your Science Classroom: Using Excel to set up a Calibration Curve

Getting to Know

Hands-on Experiments

All experiments use:

LabPro interface

Logger-Pro software

Mac Laptop and the Excel Software

Calibration Curves

A calibration curve shows the response of an analytical method to known quantities of analyte. Table-1 gives real data from a protein analysis that produces a colored product. A spectrophotometer measures the absorbance of light, which is proportional to the quantity of protein analyzed. Solutions containing known concentrations of analyte are called standard solutions. Solutions containing all the reagents and solvents used in the analysis, but no deliberately added analyte, are called blank solutions. Blanks measure the response of the analytical procedure to impurities or interfering species in the reagents.

When we scan across the three absorbance values in each row of Table-1, the number 0.392 seems out of line: It is inconsistent with the other values for 15.0 microgram, and the range of values for the 15.0-microgram samples is much bigger than the range for the other samples. The linear relation between the average values of absorbance up to the 20.0-microgram sample also indicates that the value 0.392 is in error (Figure-1). We choose to omit 0.392 from subsequent calculations

It is reasonable to ask whether all three absorbances for the 25.0-microgram samples are low for some unknown reason, because this point falls below the straight line in Figure-1. Repetition of this analysis shows that the 25.0-microgram point is consistently below the straight line and there is nothing "wrong" with the data in Table-1.

Figure-1 Average absorbance values in Table-1 versus micrograms of protein analyzed. Averages for 0 to 20 micrograms of protein lie on a straight line if the questionable datum 0.392 at 15 microgram is omitted.

Table-1 Spectrophotometer data used to construct calibration curve

Amount of

protein (microg)

Absorbance of

independent samples Range Corrected absorbance

0 0.099 0.099 0.100 0.001 -0.0033 -0.0033 0.0007

5.0 0.185 0.187 0.188 0.003 0.0857 0.0877 0.0887

10.0 0.282 0.272 0.272 0.010 0.1827 0.1727 0.1727

15.0 0.345 0.347 (0.392 ) 0.047 0.2457 0.2477 -

20.0 0.425 0.425 0.430 0.005 0.3257 0.3257 0.3307

25.0 0.483 0.488 0.496 0.013 0.3837 0.3887 0.3967

2

Constructing a Calibration Curve

The following procedure can be used for constructing a calibration curve:

Step 1 Prepare known samples of analyte covering a range of concentrations expected for unknowns. Measure the response of the analytical procedure to these standards to generate data like the left half of Table-1.

Step 2 Subtract the average absorbance (0.0993) of the blank* samples from each measured absorbance to obtain corrected absorbance. The blank measures the response of the procedure when no protein is present.

Step 3 Make a graph of corrected absorbance versus quantity of protein analyzed (Figure-2). Use the least-squares procedure to find the best straight line through the linear portion of the data, up to and including 20.0 microgram of protein (14 points, including the 3 corrected blanks, in the shaded portion of Table-1). Find the slope and intercept and uncertainties with Equations A, B, C D, E & F. The results are-

m = 0.016 30 sm = 0.000 22 sy = 0.0059

b = 0.0047 sb = 0.0026

The equation of the linear calibration line is

€

absorbancey

= m x (µg of protein)x

+b

= 90.016 30) (µg of protein) + 0.0047 Equation 1

where y is the corrected absorbance (= observed absorbance - blank absorbance).

Step 4 If you analyze an unknown solution at a future time, run a blank at the same time. Subtract the new blank absorbance from the unknown absorbance to obtain the corrected absorbance.

€

Formal Equation :y (±sy) = [m (±sm) x + [b (±sb)]

* Absorbance of the blank can arise from the color of starting reagents, reactions of impurities, and reactions of interfering species. Blank values can vary from one set of reagents to another, but corrected absorbance should not.

‡

€

Equation A:

Equation B :

Least −Square"Best" Line

Slope; m = (xiyi) ∑ xi ∑

xi ∑ n ÷ D

Intercept; b = (xi

2) ∑ (xiyi) ∑xi ∑ yi ∑

÷ D

⎧

⎨

⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪

Equation C : D = (xi

2) ∑ xi ∑xi ∑ n

Equation D: sy = (di

2) ∑n −2

Equation E :

Equation F :

Standarddeviation of slope

and intercept

sm2 =

sy2nD

sb2 =

sy2 (xi

2) ∑D

⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

3

Example Using a Linear Calibration Curve

An unknown protein sample gave an absorbance of 0.406, and a blank had an absorbance of 0.104. How many micrograms of protein are in the unknown?

Solution The corrected absorbance is 0.406 - 0.104 = 0.302, which lies on the linear portion of the calibration curve in Figure-2. Equation-1 therefore becomes

€

µg of proteins = absorbance - 0.0047

0.016 30 = 0.302 - 0.0047

0.016 30 = 18.24 µg Equation 2

Figure-2 Calibration curve for protein analysis in Table-1. The equation of the solid straight line fitting the 14 data points (open circles) from 0 to 20 microg, derived by the method of least squares, is Y = 0.01630 (+/-0.000 22) x + 0.0047 (+/-0.0026). The standard deviation of y is sy = 0.0059, The equation of the dashed

quadratic curve that fits all 17 data points from o to 25 microgram, determined by a nonlinear least squares procedure is

Y= -1.17 (+ 0.21) x 10-4 X2 + O.01B 58 (+ 0.000 46) x 0.0007 (+ 0.001

0), with Sy = 0.0046.

We prefer calibration procedures with a linear response, in which the corrected analytical signal (= signal from sample signal from blank) is proportional to the quantity of analyte. Although we try to work in the linear range, you can obtain valid results beyond the linear region ( >20 microgram) in Figure-2. The dashed curve that goes up to 25 microgram of protein comes from a least-squares fit of

the data to the equation y = ax2 + bx + c.

The linear range of an analytical method is the analyte concentration range over which response is proportional to concentration. A related quantity defined in Figure-3 is dynamic range-the concentration range over which there is a measurable response to analyte, even if the response is not linear.

Before using your calculator or computer to find the least-squares straight line, make a graph of your data. The graph gives you an opportunity to reject bad data or stimulus to repeat a measurement or to decide that a straight line is not an appropriate function. Examine your data for sensibility.

It is not reliable to extrapolate any calibration curve, linear or nonlinear, beyond the measured range of standards. Measure standards in the entire concentration range of interest.

Propagation of Uncertainty of a Calibration Curve

In the preceding example, an unknown with a corrected absorbance of y = 0.302 had a protein content of x = 18.24 microgram. What is the uncertainty in the number 18.24? A full treatment of the propagation of uncertainty gives the following results:

€

Uncertainty in x (=sx) = sy

m⎡ ⎤ 1

k+ 1

n+ (y - y)2

m2 (xi - x)2∑ Equation 3

where sy is the standard deviation of y, |m| is the absolute value of the slope, k is the

number of replicate measurements of the unknown, n is the number of data points for the calibration line (14 in Tabl-1), y is the mean value of y for the points on the calibration line, xi are the individual values of x for the points on the calibration line, and x is the

mean value of x for the points on the calibration line. For a single measurement of the unknown, k = 1 and Equation-3 gives sx = +/- 0.39 microgram. If you measure four

replicate unknowns (k = 4) and the average corrected absorbance is 0.302, the uncertainty is reduced from +/-0.39 to +/-0.23 microgram.

Figure-3

Calibration curve illustrating linear and dynamic ranges.

The confidence interval for x is + tsx, where t is Student's t for n-2 degrees of freedom. If sx = 0.23 microgram and n = 14 points

(12 degrees of freedom), the 95% confidence interval for x is + tsx = + (2.179) (0.23) = +/-0.50 microgram. To find values of t that are

not in Table-1, use the Excel function TlNV. For 12 degrees of freedom and 95% confidence, the function TlNV(0.05, 12) returns t = 2.179.

4

A Spreadsheet for Least Squares Figure-4 implements least-squares analysis, including propagation of error with Equation-3. Enter values of x and y in columns B and

C. Then select cells B10:C12. Enter the formula = LINEST(C4:C7,B4:B7,TRUE,TRUE) and press CONTROL + SHIFT + ENTER on a PC or

COMMAND⌘ + RETURN on a Mac. LINEST returns m, b, sm, sb, R2, and sy in cells Bl0:C12. Write labels in cells A10:A12 and Dl0:D12

so you know what the numbers in cells Bl0:C12 mean. Cell B14 gives the number of data points with the formula = COUNT(B4:B7). Cell Bl5 computes the mean value of y. Cell Bl6 computes the

sum S (

€

xi - x )2 that we need for Equation-3. This sum is common enough that Excel has a built in function called DEVSQ that you can find in the Statistics menu of the INSERT FUNCTION menu.

Enter the measured mean value of y for replicate measurements of the unknown in cell B18. In cell B19, enter the number of replicate measurements of the unknown. Cell B20 computes the value of x corresponding to the measured mean value of y. Cell B21 uses Equation-3 to find the uncertainty (the standard deviation) in the value of x for the unknown. If you want a confidence interval for x, multiply sx times Student's t for n-2 degrees of freedom and the desired confidence level.

We always want a graph to see if the calibration points lie on a straight line. Follow the instructions in provided by your instructor to plot the calibration data. To add a straight line, click on one data point and they will all be highlighted. Go to the CHART menu and select ADD TREND LINE. In some versions of Excel you might have to go directly to the INSERT menu and select TRENDLINE.

In the window that appears, select Linear. Go to Options in the TRENDLINE box and select Display Equation on Chart. When you click OK, the least-squares straight line and its equation appear on the graph. Double click on the line and you can adjust its thickness and appearance. Double clicking on the equation allows you to modify its format. Double click on the straight line and select Options. In the Forecast box, you can extend the trendline Forward and Backward as far as you like.

Figure-4 Spreadsheet for linear least squares analysis.

Use Excel to work on the following problems:

1. The following are spectrometric signals for methane in H2.

CH4 (Vol %) 0 0.062 0.122 0.245 0.486 0.971 1.921

Signal (mV) 9.1 47.5 95.6 193.8 387.5 812.5 1,671.9

a) Subtract the blank value (9.1) from all other values. Then use the method of least squares to find the slope and intercept and their uncertainties. Construct a calibration curve.

2. What mass of NH4 FeNH4(SO4)2•2 H2O (Solid) is required to prepare a 500mL solution containing 100 ppm in Fe?

Using this stock solution, what aliquot must be used to prepare calibration solutions, 100-mL volume, of the following 0.100 ppm, 0.500 ppm, 2.00 ppm, 4.00 ppm and 7.00 ppm.

The AA absorbance data is shown below. Solution 1 2 3 4 5

Conc Fe (ppm) 0.100 .500 2.00 4.00 7.00

Absorbance 0.0122 0.0608 0.2322 0.455 0.6058

Construct a calibration curve based on the data above and use Linest to determine: Equation of the calibration curve Slope and standard deviation, y-intercept and standard deviation, correlation factor Are there any outliers in the data? If so, recalculate the above using the new set of data.

If a blood sample contains (m/m) 0.335 % iron. Determine how much blood should be diluted in a 250mL volumetric flask with the appropriate solvent to give an absorbance reading of 0.0500.

Use your calibration cure to calculate the concentration of the iron sample.