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THINK ABOUT IT

When a toxin enters your body, it is important for a doctor to know exactly how much toxin there is. One molecule of snake venom has over 100 times as much mass as one atom of arsenic. So, 1 milligram of snake venom contains many fewer particles than 1 milligram of arsenic. To track the toxin, you need an inventory, or a count, of the atoms or molecules. However, atoms are so small that you cannot see them, let alone count them. Luckily, there are other ways of figuring out the number of atoms in a sample besides counting them one by one.

How can mass be used to count large numbers of small objects?

To answer this question, you will explore

EXPLORING THE TOPIC

Using Mass to Count

Let’s explore how to count objects by weighing them. The masses of a number of different small objects are provided in the table.

It is difficult to accurately measure the mass of an object if the object has a mass that is below the detection limit of the balance. Many electronic balances have 0.001 g as a lower weight limit. So, weighing heavier objects tends to yield more accurate results.

One way around this difficulty is to count out a collection of identical (or nearly identical) objects and weigh the collection. You can then find the mass of a single object by dividing the mass of the collection by the number of objects. This gives you an average mass of one object that is more accurate than if you had weighed the object individually.

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Once you know the average mass of a single object, you can use it to determine exactly how many objects you have in a large sample. Consider a sample of pushpins.

In addition to helping you overcome the detection limit of the balance, there is a second reason that it is important to find the average mass of the objects in a sample: There is a slight variation in the masses of the objects. For example, not all grains of rice are identical in size. Some are slightly larger and others are slightly smaller (like isotopes of atoms). So, an average mass is a more accurate predictor of the mass of a typical object.

[For a review of these math topics, see MATH Spotlight: Dimensional Analysis on page A-16 and MATH Spotlight: Accuracy, Precision, and Significant Digits on page A-1.]

PERCENT ERROR

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Chemists use percent error to express how accurate their measurements are. For example, imagine a sandwich bag contains exactly 340 beans. Suppose you calculated the number of beans using mass and came up with 352 beans. To figure out your percent error, you would use this formula:

In this case, the percent error is

Notice that saying your answer was off by 3.5% is more meaningful than saying you were off by 12 beans, which could be a lot or a little, depending on how many beans you were counting. The smaller the percent error, the more accurate your answer.

Weighing Atoms

You can apply the same method you used to count tiny objects to counting atoms. To determine the number of atoms in a 1.0 g sample, you will need to know the average mass of the atoms in the sample. The average mass of the atoms of each element is given on the periodic table in atomic mass units, or amu.

Atomic mass units are special units used for atoms because the mass of an atom in grams is so tiny. Each atomic mass unit is equal to 0.00000000000000000000000166 g. There are 23 zeros in front of the 166, making this number inconvenient to use.

Consider hydrogen atoms. Hydrogen has an average atomic mass of about 1.0 amu, or 0.00000000000000000000000166 g. How many hydrogen atoms are in a 1.0 g sample?

So 1 gram of hydrogen atoms contains 602 sextillion, or 1 mole, of hydrogen atoms.

LESSON SUMMARY

How can mass be used to count large numbers of small objects?

It is difficult to count very small objects like atoms and molecules. One method for getting a count of small objects is to weigh a large number of the objects and divide by the average weight of one object. The average atomic mass on the periodic table is equivalent to the mass of one atom of each element. This number is an average because some atoms are isotopes with different masses. The mass of a single atom is expressed in atomic mass units rather than grams. To find the number of atoms in a sample of an element, you can divide the mass of the sample by the average mass of one atom.

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