The Becker Bottle—One in a Million

Introduction

How big is a million? How long would it take to count to one million? How long would it take to measure out one million objects? Is there a shortcut for measuring out such large quantities? What does “part per million” mean? Can something that is present at a concentration of only one part per million have any real consequence? How can we measure concentrations that are as low as one part per million or even smaller?

Many science disciplines often deal with questions like these. The science world is full of very large numbers. For example, how many planets are in the universe? How many miles away is that star? Consider the enormity of a mole. We want our science students to get an idea of how much a mole actually represents. To do this, let them consider that a million is to a mole (6 x 10²³) as 1/4 teaspoon of water is to Lake Michigan! Scientists have to use scientific notation (e.g., 10²³) or special prefixes (e.g., kilo, mega) to discuss the huge numbers relative to science.

Equally difficult to comprehend are very small numbers (e.g., 10^–23). Some compounds in very small concentrations can have quite significant consequences to humans. For example, the EPA considers water unfit for human consumption if cadmium levels are above 10 ppb (0.00000001)! How can that be measured?

Concepts

• Parts per million (ppm)
• Molecular concentration

Background

This amazing 3-L bottle contains one million tiny colored spheres (actually decorative cake sprinkles, 2.5 kg of them!). The blue colored sprinkles serve as the background “solvent” particles, the yellow sprinkles have been mixed at a concentration of 10% (100,000 ppm), the red are present at 1% (10,000 ppm), white at 0.1% (1,000 ppm), pink at 0.01% (100 ppm), green at 0.001% (10 ppm) and black at 0.0001% (1 ppm). There is only one black sphere in the entire bottle. Can you find it?

The Becker Bottle is a very simple device—a bottle containing one million colored sprinkles. Yet the number of science concepts it can help demonstrate and make concrete for students is impressive. All you need is the bottle and your clever teaching style to bring the bottle to life. If you find new uses for the bottle, please let us know. 

Safety Precautions

The Becker Bottle has been sealed to prevent accidental spilling and contamination. Keep it sealed and do not allow any consumption of the sprinkles.

Procedure

Suggested Activities and Discussions 

1. Approximations

Before allowing any discussion of the bottle, hold it up, let the students look at it, and have them write down their estimate of how many sprinkles are in the bottle. You may need to walk around with the bottle to give students a closer look. Have them submit a private, written estimate. Read the estimates aloud and have students discuss how they came up with their estimations. Answers will, of course, vary considerably as will the logic used in making their estimates. Then discuss what they might have done to derive a better estimate without opening the bottle. Encourage creative thoughts about how to do estimations while considering controls for each suggestion. 

2. Uncertainty and Precision

Once the students are told that the bottle contains one million sprinkles, they will invariably ask: “Exactly one million?” To this you might answer: “What if I said ‘yes’? How would you prove me wrong?” They will answer with the obvious: “Count them.” Counting would be appropriate to test the exactness of quantities such as ten, one hundred, and maybe even one thousand. For numbers as large as one million, counting and even the idea of exactness lose meaning. For very large numbers, quantities become more like measurements. When we report the mass of an object to be 15 g, we don’t mean exactly 15 g. Even when we report it to be 15.0 g or even 15.00 g, still there is no absolute exactness associated with this. What does exactly 15 g mean? 15.0000000... on forever? How can anything ever have that specific a mass and, more importantly, how could we ever confirm it if it did? By the same token, as quantities become larger—ten thousand, one hundred thousand, one million... absolute exactness becomes less and less applicable. 

3. Counting by Weighing

Another question that arises when students are told the bottle contains one million sprinkles is: “How do you know? What did you do, count them?” To this answer: “Yes, and it was so frustrating: I got to 823,672 and then I lost count and had to start over again....” They usually don’t believe this. You might ask them how they might go about confirming that there are one million sprinkles in the bottle. You might associate this with going to the hardware store and asking the clerk for 500 6-penny nails from the bulk barrel. Would that clerk count out the nails? No! He/she would weigh them out, consulting a chart that specifies the weights for different nail types, just as a chemist who needed 2.30 x 10²² magnesium atoms for a given reaction would not count them out, but would weigh them out, consulting some kind of special chart that specifies the weights for each of the different elements (hmmm...). The periodic chart. How do biologists sample microscopic populations to determine the number of organisms in a test tube? in a lake?

Since no chart has been established for the sprinkles, the students should realize that they would have to determine how much one sprinkle weighs. How would they do this? Place one sprinkle on the scale? No, they are too light and may not all have the same mass. Count out some number onto the scale and then divide by that number. Yes. But how many...? 10? 100? 1,000? At what point is the increase in reliability not worth the extra time required? This should lead into the whole discussion on sample sizes and statistics.

4. Proportions and Order of Magnitude

Pose these questions to students:

1. If the 3-L bottle holds 1,000,000 sprinkles, how many sprinkles would it take to fill a thimble? (Assuming the thimble’s volume to be 3 mL, it could hold 1,000 sprinkles.)
2. How many would it take to fill your room? [Assuming the room to be approximately 10 m x 12 m x 2.5 m (thus V = 300 m³ or 300,000 L), it could hold 100 billion (1 x 10¹¹) sprinkles!]
3. How tall would a 100 m x 100 m rectangular tank have to be to hold one mole of sprinkles? (The tank would have to be at least 180 million km tall! That’s some tank!)

5. Concentrations and Detection

Since the Becker Bottle contains different colored sprinkles in different amounts, it can be used to illustrate any power of ten concentration from 10% (100,000 ppm) down to 0.0001% (1 ppm). The blue sprinkles serve as the background “solvent” particles. Yellow sprinkles have been mixed in at a concentration of 10% and they are so prevalent that they appear almost crowded in the bottle. Red are present at 1% and very easy to spot. White sprinkles have a concentration of 0.1% (1000 ppm), and are rather easy to find, although it takes a discerning eye to find them among all the yellow. Next come the pink sprinkles at 0.01% (100 ppm), which takes a little looking for and turning over of the bottle to bring them to the surface. Green are present at a concentration of only 0.001% (10 ppm), and they are extremely difficult to find, in part because of their scarcity and in part because they blend so easily with the blue solvent. (Note: it is not difficult to tell greens from blues, or whites from yellows, once they are found, but finding them is made harder by the fact that our eyes are not drawn to them as readily. Finally there is one black sprinkle in the bottle, representing a concentration of only 0.0001%—that’s one part per million. This black one is nearly impossible to find (although students have found it).

The frustration of trying to find the one black sprinkle can lead nicely to a discussion of how particles are detected in low concentrations and what makes some particles more difficult to detect than others. How do water test kits that detect ppm and chemical titrations help in the detection of small quantities and how are they calibrated? What if the black sprinkle had a special property, like being magnetic—would it be easier to find?

Teacher Tips

• After much use, the inside surface of the bottle may become slightly scuffed, diminishing the clarity of the plastic. Should this happen, you may simply transfer the sprinkles into a new 3 L bottle (rinsed out and allowed to dry thoroughly). If kept dry the bottle can be used for many years.

References

This bottle was created by Bob Becker, Kirkwood High School, Kirkwood, MO.