Table scientific notation

Chemists frequently work with measurements that are very large or very small. A mole, for example, contains 602,213,670,000,000,000,000,000 particles, and some analytical techniques can detect as little as 0.000000000000001 g of a compound. For simplicity, we express these measurements using scientific notation thus, a mole contains 6.0221367 X 10 particles, and the stated mass is 1 X 10 g. Sometimes it is preferable to express measurements without the exponential term, replacing it with a prefix. A mass of 1 X 10 g is the same as 1 femtogram. Table 2.3 lists other common prefixes. 

Chemists routinely measure quantities that run the gamut from very small (the size of an atom, for example) to extremely large (such as the number of particles in one mole). Nobody, not even chemists, likes dealing with scientific notation (which we cover in Chapter 1) if they don t have to. For these reasons, chemists often use a metric system prefix (a word part that goes in front of the base unit to indicate a numerical value) in lieu of scientific notation. For example, the size of the nucleus of an atom is roughly 1 nanometer across, which is a nicer way of saying 1x10- meters across. The most useful of these prefixes are in Table 2-2. 

To understand how this shorthand notation works, consider the large number 50,000,000. Mathematically this number is equal to 5 multiplied by 10 X 10X 10X 10X 10 X 10 X 10 (check this out on your calculator). We can abbreviate this chain of numbers by writing all the 10s in exponential form, which gives us the scientific notation 5 X 107. (Note that 107 is the same as lOx lOx 10x lOx 10 X 10 X 10. Table A. 1 shows the exponential form of some other large and small numbers.) Likewise, the small number 0.0005 is mathematically equal to 5 divided by 10 X 10 x 10 X 10, which is 5/104. Because dividing by a number is exactly equivalent to multiplying by the reciprocal of that number, 5/104 can be written in the form 5 X 10-4, and so in scientific notation 0.0005 becomes 5 X 10-4 (note the negative exponent). 

Table A.2 shows scientific notation used to express some of the physical data often used in science.

Note that the characteristic is determined by the power to which 10 is raised (when the number is in standard scientific notation), and the mantissa is determined by the log of the lefthand factor (when the number is in scientific notation). It is these properties that make it so easy to find the logarithm of a number using a log table. Here is how you can do it. 

Look up the mantissa in the log table. It is the log of the lefthand factor in scientific notation, which is a number between 1 and 10. The mantissa will lie between 0 and 1. 

Table 1.2 Names and abbreviations used to specify the order of magnitude of numbers expressed in scientific notation...

Notice how numbers that are either very large or very small are indicated in Table 1.4 using an exponential format called scientific notation. For example, the number 55,000 is written in scientific notation as 5.5 X 104, and the number 0.003 20 as 3.20 X 10 3. Review Appendix A if you are uncomfortable with scientific notation or if you need to brush up on how to do mathematical manipulations on numbers with exponents. 

Table B.l. Chemical reactions and their equation parameters used in the FREZCHEM model (version 9.2).a (Numbers are in computer scientific notation, where e xx stands for 10 xx)... 

Table B.8. Equations for the molar volumes (cm3/mole) and the isothermal compressibilities [cm3/(mole.bar)] of soluble ions and gases at infinite dilution. Derived from the database of Millero (1983) over a temperature range of 273 to 298 K. (Numbers are in computer scientific notation, where e xx stands for 10 a a ). Reprinted from Marion et al. (2005) with permission...

The metric system is a decimal system, based on powers of 10. Table 2.5 is a list of the prefixes for the various powers of 10. Between scientific notation and the prefixes shown below, it is very simple to identify, name, read, and understand 36 decades of power of any given base or derived unit. 

Safety of chemicals. Scientific notation, conversions Periodic Table of Elements Naming simple compounds Atomic structure and periodicity Balancing equations Drawing molecules Interaction of light with molecules Chlorofluorocarbons (CFC s) and ozone Development of green pesticides... 

Time The SI base unit for time is the second (s). The frequency of microwave radiation given off by a cesium-133 atom is the physical standard used to establish the length of a second. Cesium clocks are more reliable than the clocks and stopwatches that you use to measure time. For ordinary tasks, a second is a short amount of time. Many chemical reactions take place in less than a second. To better describe the range of possible measurements, scientists add prefixes to the base units. This task is made easier because the metric system is a decimal system. The prefixes in Table 2-2 are based on multiples, or factors, of ten. These prefixes can be used with all SI units. In Section 2.2, you will learn to express quantities such as 0.000 000 015 s in scientific notation, which also is based on multiples of ten. 

Recall from Table 4-1 that the masses of both protons and neutrons are approximately 1.67 x 10 g. While this is a very small mass, the mass of an electron is even smaller—only about that of a proton or neutron. Because these extremely small masses expressed in scientific notation are difficult to work with, chemists have developed a method of measuring the mass of an atom relative to the mass of a specifically chosen atomic standard. That standard is the carbon-12 atom. Scientists assigned the carbon-12 atom a mass of exactly 12 atomic mass units. Thus, one atomic mass unit (amu) is defined as the mass of a carbon-12 atom. Although a mass of 1 amu is very nearly equal to the mass of a single proton or a single neutron, it is important to realize that the values are slightly different. As a result, the mass of silicon-30, for example, is 29.974 amu, and not 30 amu. Table 4-2 gives the masses of the subatomic particles in terms of amu. 

Rewrite the following in scientific notation, using only the base units of Table B.l, without prefixes. 

Finally, the nvalue column of the vla4.property table can be populated when possible. This column stores the numerical value of the property. Since not all values are numerical, this column may have null entries. The purpose is to enable efficient use of numerical data when appropriate, for example, to select by value, sort, apply mathematical functions, etc. The following SQL will update the nvalue column when possible with a numeric value. The tilde operator in the where selects text values that match the regular expression. The expression shown here allows integers, decimal values, and scientific notation using E or e for the exponent, for example 6.023E23. 

There are two great advantages to scientific notation. The first, as illustrated in Table B.l, is that the very large and very small numbers often dealt with in the sciences are much less cumbersome in scientific notation. The second is that it removes the ambiguity in the number of significant figures in a number ending with zeroes. We can, for example, write... 

The advantage of the SI system is that it is a measuring system based on a decimal system. With calculations written in groups of ten, results can be easily recorded as something called scientific notation. There are written prefixes that indicate exponential values as well. Some of these are listed in Table 2.2 which lists terms used in scientific notation. 

Table 2.2 Scientific notation helps determine the scale of measurements.

Multiplication and division calculations involving scientific notation are easily done using a hand calculator, t Table 1.4 gives the steps, the typical calculator procedures (buttons to press), and typical calculator readout or display for the division of 7.2 X 10 by 1.2 X 10 . 

The SI system is based on seven fundamental units, or base units, each identified with a physical quantity (Table 1.1). All other units are derived units, combinations of the seven base units. For example, the derived unit for speed, meters per second (m/s), is the base unit for length (m) divided by the base unit for time (s). (Derived units that are a ratio of base units can be used as conversion factors.) For quantities much smaller or larger than the base unit, we use decimal prefixes and exponential (scientific) notation (Table 1.2). (If you need a review of exponential notation, see Appendix A.) Because the prefixes are based on powers of 10, SI units are easier to use in calculations than English units. 

The amount of significant figures in a number is perhaps determined easiest from the scientific notation for the number. Some examples are listed in Table C.l. 

Table 2.2 gives examples of numbers written as positive and negative powers of 10. The powers of 10 are a way of keeping track of the decimal point in the decimal number. Table 2.3 gives several examples of writing measurements in scientific notation. 

TABLE 2.3 Some Measurements Written in Scientific Notation... 

Scientific notation (see Appendix lA) allows us to express very large or very small quantities in a compact manner by using exponents. For example, the diameter of a hydrogen atom can be written as 1.06 x 10 m. The International System of Units uses the prefix multipliers shown in Table 1.2 with the standard units. These multipliers change the value of the unit by powers of 10 (just like an exponent does in scientific notation). For example, the kilometer has the prefix kilo meaning 1000 or 10. Therefore,... 

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