As we have learned in math courses, exponents of ten can be used to express numbers and simplify calculations. Let’s look at a further use of exponents which is better suited to measurements and calculations in chemistry.
A number such as 125 can be expressed as 102.0969, but a much more straightforward way of expressing this number would be as 1.25 x 100 or as in exponential notation 1.25 x 102. Or instead of expressing 13,000 as 104.1139 we write 1.3 x 1000 or as in exponential notation 1.3 x 103. This notation is called scientific notation and consists of a coefficient, the number to the left of the multiplication sign, and an exponential, ten raised to a whole number exponent. Using this method negates having to fumble around with long cumbersome logarithms. The coefficient in scientific notation may contain a number of any length, but only a number between one and ten can be placed to the left of the decimal point. For example, to express the number 2,341 in scientific notation we first develop the coefficient by placing a decimal point directly after the two: 2.341
Second, determine what number must be used to multiply 2.341 to give the proper magnitude. In this case, the number is 1,000.
Using the exponential form of 1000, 103 results in proper scientific notation of:
Another way of looking at this is to observe that when multiplying by 10 to a power, the decimal point is moved the same number of places as the exponent. For our example, 2,341, the decimal point is moved to the right of 2 which is our first nonzero number. The decimal point has been moved 3 places to the left of the implied decimal point.
Small numbers can also be put into scientific notation. To express the number 0.0000315 in scientific notation, we develop the coefficient by placing a decimal point directly after the first nonzero number
Second, determine what exponential must be used to multiply 3.15 to give the proper magnitude. In this case, the exponential is 10-5. The resulting scientific notation is
Note that we move the decimal point 5 places to the right to obtain 10-5. When converting numbers to scientific notation if the decimal point is moved to the left, the exponent is positive, and if it is moved to the right, the exponent is negative.
You will use scientific notation in most of your chemistry courses. Please make sure you can write both small and large numbers and measurements in scientific notation. Students should be able to convert between standard and scientific notation.
1. Express the following in scientific notation
a) a mass of 45,000 pounds
b) the diameter of a chloride atom (0.000000000145 m)
c) the distance from the Earth to Mars (78,340,000 km)
2. Convert the following to standard notation
a) 6.7200 x 108m
b) 4.6800 x 10-4L