CHEM 2204

Chemical Laboratory Techniques

Browsing Posts in Graphing

As part of your lab writeup for Lab 5, you will need to hand in three properly labeled graphs:

Graph 1: Absorbance versus Wavelength – Indicate λmax and its value on the graph [Graph the fine scan only]
Graph 2: %Transmittance versus Wavelength – Indicate λmax and its value on the graph [Graph the fine scan only]
Graph 3: Absorbance versus Concentration – Run Linest on the data – Indicate the unknown sample on the graph by dropping a line from the absorbance of the sample to the x-axis. Indicate the concentration of the dilute sample and the original sample.

Read the detailed tips on how to plot these graphs.

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For Graph 1 and 2, after the data points are on the graph, right mouse click over one of the points and choose Change Series Chart Type … from the context menu. In the Change Chart Type popup, under (X,Y Scatter), choose Scatter with Smooth Lines and Markers.

This will draw a smooth curve through the data points.

Now add minor gridlines to the y-axis by right mouse click on the y-axis.

Choose Add Minor Gridlines. This will draw the minor gridlines for the y-axis.

Now add minor gridlines to the x-axis by right mouse click on the x-axis by right mouse click on the x-axis.

Choose Add Minor Gridlines. This will draw the minor gridlines for the x-axis.

With the minor axes drawn, this will allow you to interpolate the λmax value from the graph accurately.
Be sure to indicate λmax and its value on Graph 1 and 2.

For Graph 3, when you add the Linest function to the graph, be sure to enter false for Const. This will force the line to go through (0,0).

Be sure to indicate the unknown and its value on Graph 3.

Based on the following set of data,

the graph generated would be

For this graph that you are generating for next week, keep all the digits.

  1. You need to convert all the temperature readings to Kelvin, so add 273.15. (Keep all digits)
  2. You need to take the inverse of temperature readings that have been converted to Kevin. (Keep all digits)
  3. You need to take the natural log (ln) of all the pressure readings. (Keep all digits)
  4. Add the trendline and R2 value on the graph.

Make sure your graph is properly labeled with all the following information:

  1. Title the graph
  2. Label x-axis
  3. Label y-axis
  4. Write the slope = ________ units
  5. Calculate ΔHvap = _________ units

Run LINEST to determine the number of significant figures for the slope and ΔHvap.

From page 1, we learned that if we believe the temperatures were poorly measured, it would lead to the determination of ΔHvap of water to 2 significant figures.

But the graph of the data shows a very good fit of the data to a straight line ( R2= 0.998), so maybe the temperature measurements were more exact than how the data was recorded. Let’s determine the statistical error on the data using Excel’s LINEST function …

The LINEST function calculates the statistics for a line by using the “least squares” method to calculate a straight line that best fits the data, and then returns an array that describes the line.

Follow these steps:
1. LINEST function returns several outputs, so to begin, select a 2 by 5 array as shown.
See screen capture.

2. Click , the Insert function icon.
See screen capture

3. In the Insert Function popup menu, choose the Statistical category and then choose the LINEST function. Click OK.
See screen capture

4. In the Function Arguments window, click the Known y’s field and drag your mouse to highlight the y-values.
See screen capture

5. Click into the Known_x’s field and drag your mouse to highlight the x-values.
See screen capture

6. Type TRUE in the Const and Stats fields and click OK.
See screen capture

7. Highlight the formula in the formula bar.
See screen capture

8. Press Ctrl+Shift simultaneously and hit Enter. The array we selected in Step 1 is now filled with numbers.
See screen capture

9. Let’s focus on the meaning of the highlighted numbers.

The LINEST function calculates the statistical errors of the slopes and intercepts.

Slope = -5090 +/- 50 K (3 significant figures)
Intercept = 18.2 +/- 0.2 (3 significant figures)

From the slope, we determine ΔHvap of water.

slope = – ΔHvap / R where R is the gas constant, 8.314 J·mole-1·K-1.

ΔHvap = (5090)(8.314) = 42.3 kJ/mole (3 significant figures)
(Note: If you use the slope value of 5086.96266 to calculate ΔHvap, your answer, when rounded to 3 significant figures, should also give you 42.3 kJ/mole)

This is how one determines the uncertainty in the slope and intercept of a graph using LINEST.

In summary, the statistical uncertainty determined by LINEST is the better representation of the uncertainty in this set of data. Had the coefficient of determination for the data not been close to 1, meaning that the data does not show a linear relationship, then one cannot simply determine the uncertainty of the slope and intercept based on statistics alone.

Take a look at the two-column data.

T (oC) P (kPa)
0 0.612
10 1.227
20 2.536
30 4.242
40 7.37
50 12.33
60 19.9
70 31.15
75.7 46.12
89.7 70.1
100 101.32
120 198.5
200 1554.3

If we look closely at the data and pay attention to where the uncertainties are, we can conclude that the temperature readings have huge uncertainties. Are they real uncertainties, or are they sloppily recorded without consideration of significant figures? We could ask the person who took the data for clarification, but let’s assume we don’t know who took the data, and we have to proceed with our graphing exercise because it is due next Friday!

Okay, the due date is near and we can’t wait; we will just have to assume that the data was collected with a temperature measuring device that is not very good. According to significant figure rules, the uncertainties in the temperature readings are as follows:

Temperature
(oC)
Uncertainty in
temperature readings
Range in temperature
(oC)
0 1 sig fig; ± 0.5o -0.5 – 0.5
10 1 sig fig; ± 5oC 5 -15
20 1 sig fig; ± 5oC 15 -25
30 1 sig fig; ± 5oC 25 -35
40 1 sig fig; ± 5o 35 -45
50 1 sig fig; ± 5oC 45 -55
60 1 sig fig; ± 5oC 55 -65
70 1 sig fig; ± 5oC 65 -75
75.7 3 sig figs; ± 0.05oC 75.65 -75.75
89.7 3 sig figs; ± 0.05oC 89.65 -89.75
100 1 sig fig; ± 50oC 50 -150
120 1 sig fig; ± 5oC 115 -125
200 1 sig fig; ± 50oC 150 -250

We keep these uncertainties in mind and go to Excel to plot some graphs!

Graph 1: Use the data as is and plot “ln Pressure Vs. 1/Temperature”. Here is the graph.

Graph 2: Use the upper error limit on temperature and plot “ln Pressure Vs. 1/Temperature”. Here is the graph.

Graph 3: Use the lower error limit on temperature and plot “ln Pressure Vs. 1/Temperature”. Here is the graph.

Graph 2 and 3 illustrate the worst case scenario where maximum error could be observed. The trend lines for the three graphs are summarized below.

Graph 1 : Trendline equation is y = -5085x + 18.23 (Best line)
Graph 2 : Trendline equation is y = -4501x + 16.16
Graph 3 : Trendline equation is y = -5353x + 19.6

This gives us a +/- uncertaintiesfor the slope (approximately +300 and -600) and intercept (approximately +2 and -3):

  • The uncertainty in the slope is in the hundreds digit, therefore the slope would be -5100 K, with 2 significant figures.
  • The uncertainty in the intercept is in the ones digit, therefore the intercept would be 18, with 2 significant figures.

From the slope, we determine ΔHvap of water.

slope = ΔHvap / R where R is the gas constant, 8.314 J/moleK.

ΔHvap = (5100)(8.314) = 42 kJ/mole (2 significant figures)

The above ΔHvap is calculated assuming maximum errors in the temperature measurements.

Since Graph 1 shows a very good fit of the data to a straight line ( R2= 0.998), perhaps the temperature measurements were more exact than how the data was recorded. Let’s determine the statistical error of the data using Excel’s LINEST function … (see next post)