When comparing atoms to larger objects -- with a large disparity in size -- orders of magnitude show how to quantify the size differences. Orders of magnitude allow you to compare the approximate value of an extremely small object, such as the mass or diameter of an atom, to a much larger object. You can determine the order of magnitude using scientific notation to express these measurements and quantify the differences.
TL;DR (Too Long; Didn't Read)
To compare the size of a large atom to a much smaller atom, the orders of magnitude allow you to quantify the size differences. Scientific notations help you to express these measurements and assign a value to the differences.
The Tiny Size of Atoms
The average diameter of an atom is 0.1 to 0.5 nanometers. One meter contains 1,000,000,000 nanometers. Smaller units, such as centimeters and millimeters, typically used to measure small objects that can fit within your hand, are still much larger than a nanometer. To carry this further, there are 1,000,000 nanometers in a millimeter and 10,000,000 nanometers in a centimeter. Researchers sometimes measure atoms in ansgtoms, a unit that equals 10 nanometers. The size range of atoms is 1 to 5 angstroms. One angstrom equals 1/10,000,000 or 0.0000000001 m.
Units and Scale
The metric system makes it easy to convert between units because it is based on powers of 10. Each power of 10 is equal to one order of magnitude. Some of the more common units for measuring length or distance include:
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- Kilometer = 1000 m = 103 m
- Meter = 1 m = 101 m
- Centimeter = 1/100 m = 0.01 m = 10-2 m
- Millimeter = 1/1000 m = 0.001 m = 10-3 m
- Micrometer = 1/1,000,000 m = 0.000001 m = 10-6 m
- Nanometer = 1/1,000,000,000 m = 0.000000001 m = 10-9 m
- Angstrom = 1/10,000,000,000 m = 0.00000000001 m = 10-10 m
Powers of 10 and Scientific Notation
Express powers of 10 using scientific notation, where a number, such as a, is multiplied by 10 raised by an exponent, n. Scientific notation uses the exponential powers of 10, where the exponent is an integer that represents the number of zeros or decimal places in a value, such as: a x 10n
The exponent makes large numbers with a lengthy series of zeros or small numbers with many decimal places much more manageable. After measuring two objects of vastly different sizes with the same unit, express the measurements in scientific notation to make it easier to compare them by determining the order of magnitude between the two numbers. Calculate the order of magnitude between two values by subtracting the difference between its two exponents.
For example, the diameter of a grain of salt measures 1 mm and a baseball measures 10 cm. When converted to meters and expressed in scientific notation, you can easily compare the measurements. The grain of salt measures 1 x 10-3 m and the baseball measures 1 x 10-1 m. Subtracting -1 from -3 results in an order of magnitude of -2. The grain of salt is two orders of magnitude smaller than the baseball.
Comparing Atoms with Larger Objects
Comparing the size of an atom to objects large enough to see without a microscope requires much greater orders of magnitude. Suppose you compare an atom that has a diameter of 0.1 nm with a size AAA battery that has a diameter of 1 cm. Converting both units to meters and using scientific notation, express the measurements as 10-10 m and 10-1 m, respectively. To find the difference in the orders of magnitude, subtract the exponent -10 from the exponent -1. The order of magnitude is -9, so the diameter of the atom is nine orders of magnitude smaller than the battery. In other words, one billion atoms could line up across the diameter of the battery.
The thickness of a sheet of paper is about 100,000 nanometers or 105 nm. A sheet of paper is about six orders of magnitude thicker than an atom. In this example, a stack of 1,000,000 atoms would be the same thickness as sheet of paper.
Using aluminum as a specific example, an aluminum atom has a diameter of about 0.18 nm compared with a dime that has a diameter of about 18 mm. The diameter of the dime is eight orders of magnitude greater than the aluminum atom.
Blue Whales to Honeybees
For perspective, compare the masses of two objects that can be observed without a microscope and are also separated by several orders of magnitude, such as the mass of a blue whale and a honeybee. A blue whale weighs about 100 metric tons, or 108 grams. A honeybee weighs about 100 mg, or 10-1 g. The whale is nine orders of magnitude more massive than the honeybee. One billion honeybees have about the same mass as one blue whale.