2: Dilutions and Densities

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Dilutions

Learning Objectives and Skills

Students will
Students will

Background Information

Last week, we found that it is often easier to precisely measure mass rather than volume. However, when very small masses are needed in relatively large volumes, we often cannot measure small enough masses without using very large volumes. For example, we often measure metals in solutions in units like parts per million, meaning that if we wanted 1.00 grams of zinc in a solution, we would need to have a million grams of solution. For water, this would be about 1000 liters (L). It would be very wasteful to prepare solutions this way, so instead, solutions are prepared and then diluted to various concentrations as needed. This lab will focus on working through dimensional analysis for these dilutions to discover the most important equation for dilutions, and probably the most common equation used by research chemists and cellular biologists. We will be using a compound that absorbs light linearly with concentration, allowing us to quickly see if the dilution went well.

The dilution in this lab is a 1000-fold dilution meaning that the final solution's concentration is one one-thousandth (1/1000) of the original concentration. Thus, it cannot be done accurately in a single step. A more accurate way is to dilute your starting solution part way and then dilute this new solution again. Several of these steps can be used to make extreme dilutions from a starting solution and still be very accurate. This method is called a serial dilution because each new dilution is subsequently diluted again, making increasingly more dilute solutions.

Equipment and Materials

Balance
Weighing paper
0.1 grams of fluorescein
Sodium carbonate
50 mL beaker
25 mL volumetric flask
10 mL and 50 mL graduated cylinders
Deionized water
Glass stirring rod
Beral pipet
Parafilm®
LabQuest and SpectroVis instrument
6 plastic cuvettes
Micropipette
Microsoft Excel document (downloaded and saved to your computer before you start)

Procedure

Step 1 (with partner)

1.1. Using a balance and weighing paper, obtain about 0.1 (±0.005) g of the reddish colored powder (fluorescein) from the vial next to the balance. Record the actual mass with all available significant figures. Be as careful and clean as possible. Immediately add the powder to a 50 mL beaker.
1.2. Rinse all powder into the beaker with a small amount of deionized (DI) water from a water bottle. Add about 10 mL of 0.2 M $\ce{Na2CO3}$ base to this solid and dissolve the solid by carefully stirring with a glass stirring rod.
1.3. Once dissolved, quantitatively transfer (i.e., transfer it all by rinsing with multiple small volumes) this solution into a clean 25 mL volumetric flask by successively washing the beaker with small amounts (3-5 mL) of DI water, pouring each rinse into the flask being sure to stay under the 25 mL mark. Use a beral pipet to accurately reach the 25 mL mark. If you go over the 25 mL mark, you must do step 1 and 2 over. Seal the volumetric flask with Parafilm® and mix well by inverting and swirling several times.

Step 1 Questions:

1. What is the actual mass, including all significant figures and units, of your fluorescein?
2. What is the actual volume, including an appropriate number of significant figures and units, of your fluorescein? This should include the decimal place indicated by the tolerance of the flask. For example, a 25 mL volumetric flask with a tolerance of ±0.1 mL should be reported as 25.0 mL, while a 25 mL volumetric flask with a tolerance of ±0.01 mL is reported as 25.00 mL.
3. Do you think that your mass reading or your volume reading introduces a larger degree of uncertainty (less precision)? Why?
4. Concentration units are often given in a mass per volume or an amount per volume. What is the concentration of the fluorescein solution you made in these steps? Include the appropriate number of significant figures and units.

Step 2 (with partner)

Note: Be very accurate in this series of steps. 5.0 mL of a solution plus adding enough DI water to reach a total of 50.0 mL yields a 10-fold dilution because the volume has increased 10-fold while the amount in the 5.0 mL has not changed.

2.1. Obtain 5.0 mL of your solution using your 10 mL graduated cylinder and quantitatively transfer it to your 50 mL graduated cylinder with several small washes of DI water, bringing the total volume up to 50.0 mL. Seal your 50 mL graduated cylinder with Parafilm® and invert several times to mix. You now have made a 10-fold dilution of your original solution. Place this solution in a clean beaker.
2.2. Repeat step 2.1 with this exception: Measure 5.0 mL of your 10-fold diluted solution (what is now in your beaker) instead of your original solution. Add DI water to bring the total volume up to 50.0 mL. This will give a 100-fold dilution of your original solution (10-fold times 10-fold). This should be visibly more dilute than the previous solution.
2.3. Repeat step 2.1 a third time, this time using 5.0 mL of the 100-fold diluted solution. Add DI water to bring the total volume up to 50.0 mL. After the final mixing, you will have a 1000-fold diluted solution. As can be seen, this sequence of three steps (2.1-2.3) is called a serial dilution since each solution is further diluted by the next consecutive step in the sequence.

Step 2 Questions:

5. What is the mass of fluorescein represented by the 5.0 mL sample in step 2.1? Use the 5.0 mL volume, your concentration amount from question 4, and dimensional analysis. Include the appropriate number of significant figures and units.
6. After bringing the volume up to 50.0 mL in step 2.1, does the mass of fluorescein present change? Why or why not?
7. Using the same concentration units you used in question 4, what is the new concentration of fluorescein at the end of step 2.1? At the end of step 2.2? At the end of step 2.3? Make sure to include the appropriate number of significant figures and units for each answer.
8. Notice the general form of your calculations. As you dilute from initial concentration, $\ce{C1}$, with a volume, $\ce{V1}$, to a new volume, $\ce{V2}$, you arrive at a new concentration, $\ce{C2}$. The calculation you performed by dimensional analysis is: \begin{align*} \large \ce{\frac{C1$*$V1}{V2} = C2}\qquad or \qquad \ce{C1$*$V1 = C2$*$V2} \end{align*}The dimensional analysis always works out if the units for concentration and volume are the same for each value. What would the final concentration units need to be if your initial concentration was in mg / dL?

Step 3 (with partner)

In this step, you will be obtaining the absorbance of the final solution (the most diluted made in step 2.3) with the LabQuest and SpectroVis instrument.

3.1. Use the USB cable to connect the SpectroVis (Spec) to the LabQuest and then turn on the LabQuest with the power button in the top-left corner. Obtain a plastic cuvette (do not touch its clear sides; only its frosted sides), fill it about ¾ full with DI water, and carefully place in the Spec so that the clear sides are lined up with the light and detector.
3.2. Calibrate the SpectroVis by selecting the following menus/buttons with the stylus (NEVER A PEN OR PENCIL):
(Sensors / Calibrate / USB:Spectrometer / {wait for the warm up period} / Finish Calibration / OK).
3.3. Remove and empty the cuvette, fill it about ¾ full [How much is this - 3.0 mL?] with your 1000-fold dilution (step 2.3), and place in the Spec with the clear sides lined up with the light and detector. Start data collection by tapping the green arrow button in the lower left corner of the screen (select discard if asked) and then the red stop button (located in the same spot as the green arrow turns into a red square).
3.4. Select the wavelength with the highest absorbance (using the left and right arrow keys if needed). If it is not close to 490 nanometers (nm), consult your instructor. Record this wavelength and the absorbance value (Abs) given. Show these values and the spectrum to your instructor for validation.
3.5. Label a white piece of paper with numbers 1-6, spaced wide enough that each number could have a cuvette placed on top. Place the cuvette from step 3.3 (the one presently inside the Spec) over a piece of paper labeled "1". Using a calibrated micropipette, create 5 additional cuvettes by using the chart to add the following amounts of deionized water and your most dilute fluorescein solution (step 2.3). Place the filled cuvettes (each now with 3.0 mL of combined solution) on the corresponding number on your paper.

Cuvette	2	3	4	5	6
Deionized Water (mL)	0.5	1.0	1.5	2.0	2.5
Fluorescein (mL)	2.5	2.0	1.5	1.0	0.5

Step 3 Questions:

9. Based on your LabQuest spectrum, why is it important to specify a wavelength when working with light absorption?
10. When looking at your solution from step 2.3 in the cylinder and the solution of the same concentration in cuvette 1, does one appear to absorb more light than the other? If they are the same concentration, why do you think this is?
11. Look at the 6 cuvettes you created in step 3.5. Which is most concentrated? How does the color change with concentration?
12. As you diluted your fluorescein solutions, what happened to the amount of transmitted light through the solution? Was transmission higher or lower at high concentrations?
13. How would you express the amount of light transmitted as a percent of the maximum possible light transmitted, using the variables from Figure 1?

Definition: Absorbance

Concentrations are difficult to measure directly. You could, in theory, evaporate all the solvent off and measure the remaining mass. That would work if you only had one dissolved component in a mixture. In most cases, we take advantage of relative consistency in absorptivity for compounds to establish standardized controls that relate concentration and absorbance.

Absorbance is a calculated value based on what we can actually measure – the intensity or power of light. Figure 1 shows how the power of light detected changes as the sample absorbs some portion of light.

Schematic diagram showing the attenuation of radiation passing through a sample.

Figure $\PageIndex{1}$: 1a) Incident power (P₀) is diminished to transmitted power (P_T) as it passes through a sample to a detector. 1b) Incident power and transmitted power are the same (P0) when only a blank sample is present. (From: Analytical Chemistry 2.1 by David Harvey (Summer 2016) via (CC BY-NC-SA 4.0))

Step 4 (individual Excel work with question)

Note: Download a copy of the linked Excel spreadsheet and save your own local copy. Open this local copy in Excel and switch to the sheet entitled "Beer’s Law". A table has been created and partially completed on this sheet.

4.1. When you have data that varies predictably, Excel can auto-fill the data. Click and drag from the middle of Cell B4 to Cell C5 to highlight a 2x2 block of cells. Hover over the bottom right corner so that your cursor becomes a large, all-black cross. Click and drag the cross to column G. Verify that the amounts indicated match those from the table/chart in step 3.5.
4.2. Enter the concentration of fluorescein you obtained at the end of step 2.3 (see question 8) into cell B6 (make sure your units are correct!). Notice that the cell to the right automatically updated, as it was filled in with a formula. Click on this cell (C6) to view the formula. Near the top of the window, the function should be visible as =$B$6*C5/(C4+C5). Cell B6 is the concentration of the fluorescein solutions you are using for every one of the dilutions ($\ce{C1}$). Cell C5 is the volume of the fluorescein solution ($\ce{V1}$) you are diluting to the final volume ($\ce{V2}$), which is the sum of the volume of fluorescein (C5) and the volume of water (C4) used to dilute it.
4.3./Question 14. Click in the bottom right of cell C6 to create the cross and drag to cell G6. What is the formula now listed in cell G6? What did placing $ in front of the row and column labels for cell B6 do when copying the formula to adjacent cells?

Step 5 (with partner [5.1-5.2] and individual)

5.1 On the graph screen of the LabQuest, make sure the wavelength of maximum absorbance is still selected. Return to the meter screen using the meter icon in the top banner and set the units for the spectrophotometer to % transmittance (%T) by clicking on the red banner. Select “Change Units” and “% Transmittance”.
5.2. Input each cuvette (1-6) into the spectrophotometer to measure the % transmittance, carefully returning each to your sheet after you are finished. Record these values in Row 7 of your local copy of the linked Excel spreadsheet.
5.3. Absorbance is a calculated value defined as the negative log of transmittance (what isn’t transmitted is absorbed). Since transmittance is often reported as a percent, this needs to be corrected in the calculation by dividing % Transmittance by 100 to express as a decimal. This calculation is completed in cell B8. Use the cross in the bottom right corner to drag this calculation over to extend it through G8.
5.4. A graph showing % transmittance (%T) vs. concentration should have automatically been filled out. Next you will create your own graph relating Absorbance vs. Concentration.
5.5. Navigate to the insert menu of Excel. Click on the charts submenu and select "Scatter". It should look like this icon:
5.6. Click and drag the blank graph window so that it is below the %T vs. Concentration graph.
5.7. Excel may default to “Chart Design” when the chart is selected. For older versions of Excel, you may have to click in the blank chart and select "Chart Filters". Regardless of what version of Excel you are using, you are looking to choose “Select Data”. You may also be able to right click on the chart and choose “Select Data” as well.
5.8. Under “Legend Entries (Series)” on the left, click the “Add” button.
5.9. Type “Absorbance vs. Concentration” under “Series name:”.
5.10. Click the arrow next to “Series X values:”. Click and drag over your concentration values (Cells B6-G6), which will be plotted against the x-axis. Hit enter or return to validate the selection.
5.11. Click the arrow next to “Series Y values:”. Click and drag over your Absorbance values (Cells B8-G8), which will be plotted against the y-axis. Hit enter or return to validate the selection. Hit OK once. Before you hit OK a second time to close the window, ensure that only your new series (Absorbance vs. Concentration) is selected.

Step 5 Questions:

15. As you filled out the spreadsheet, a graph relating % transmittance (%T) vs. concentration should have automatically filled in your data. Does your measured relationship between % transmittance and concentration match what you observed with your eye in question 11?? Is it a linear relationship?
16. In steps 5.5-5.11, you created a chart plotting absorbance vs. concentration. Is your plot of Absorbance vs Concentration linear? Does it match the plot in the %T vs. concentration chart?

Further Background Information

You should have a relatively straight line for absorbance, but not for % transmittance. Earlier you saw that the amount of light absorbed is related to 1) the concentration of the sample, 2) the length of the path through the sample that the light travels (Question 11?), and 3) the wavelength of light being used. But a fourth component is the intensity of the light itself. As the light travels through the sample, it is being absorbed, which in turn reduces the amount of light that can be absorbed as the light continues to travel through more of the sample. This constant light loss increases the complexity of light absorption, and we must use calculus to get an accurate equation describing the absorbing process. Using calculus, we find that the integral of dP/P must be taken, which introduces a log function resulting in the following equation:

\[ \large -\log \left(\frac{P}{P_0}\right)=\varepsilon * \ell * C\]

where $ \large \varepsilon$ (the molar absorptivity) is a constant for each wavelength for each compound, $ \large \ell$ is the path length that the light travels through the sample (our cuvettes have a path length of 1 cm), and $ C$ is the concentration of the compound that absorbs the light. This $ -\log \left(\frac{P}{P_0}\right) $ is also called the Absorbance (which is dependent upon the wavelength of light) and through substitution we can write:

\[ \large A b s=\varepsilon * \ell * C\]

This is the Beer-Lambert Law, and is a powerful way to measure concentrations of solutions.

Step 5 Questions (continued):

17. Click on any of the data points displayed on your created Absorbance vs. Concentration graph to select the data series, then right click and select “Add Trendline”. An options window or menu should appear, with a “Linear” trendline set as the default. Scroll down in this menu to check “Display Equation on chart” and “Display R-squared value on the chart”. The equation will be in the form $ y =mx + b$, where $ y$ is the Absorbance and $ x$ is the concentration. The $ b$, the x-intercept, should be very close to zero indicating little to no absorbance for the blank. If the $ b$ value is zero, this $ m$ value (the slope) should be equal to the $ \large \varepsilon$ (molar absorptivity) * $ \large \ell$ (path length) from the Beer-Lambert Law. Given the path length is 1 cm, what is the molar absorptivity with units of the fluorescein based on the slope of your trendline?

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