Measurements in scientific notation

Be a good chemist. Report your measurements in scientific notation to avoid such annoying ambiguities. (See the earlier section Using Exponential and Scientific Notation to Report Measurements for details on scientific notation.)... 

Suppose you must do a calculation using the measurement 200 L. You cannot be certain which zero was estimated. To indicate the significance of digits, especially zeros, write measurements in scientific notation. In scientific notation, all digits in the decimal portion are significant. Which of the following measurements is most precise ... 

Expressing small measurements in scientific notation is done in a similar way. The diameter of a carbon atom is 0.000 000 000 000 154 m. In this case, you move the decimal point right imtil it is just past the first nonzero digit—in this case, 1. The number of places you move the decimal point right is expressed as a negative exponent of ten. The diameter of a carbon atom is 1.54 X 10 m. You always move the decimal point imtil the coefficient of ten is between one and less than ten. Thus, scientific notation always has the form, M X 10" where 1 < M < 10. 

Write each of the following measurements in scientific notation. 

Adding and Subtracting Measurements in Scientific Notation Adding and subtracting measurements in scientific notation requires that for any problem, the measurements must be expressed as the same power of ten. For example, in the following problem, the three length measurements must be expressed in the same power of ten. 

In adding and subtracting measurements in scientific notation, all measurements are expressed in the same order of magnitude as the measurement with the greatest power of ten. When converting a quantity, the decimal point is moved one place to the left for each increase in power of ten. 

Multiplying and Dividing Measurements in Scientific Notation Multiplying and dividing measurements in scientific notation requires that similar operations are done to the numerical values, the powers of ten, and the emits of the measurements. 

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. 

Any zero to the right of nonzero digits and to the left of a decimal point and not covered by rule 2 may or may not be significant, depending on whether the zero is a placeholder or actually part of the measurement Such a number should be expressed in scientific notation to avoid any confusion. 

When you know how to express your numbers in scientific notation and how to distinguish between precision and accuracy (we cover both topics earlier in this chapter), you can bask in the glory of a new skill using scientific notation to express precision. The beauty of this system is that simply by looking at a measurement, you know just how precise that measurement is. 

Mole A mole is a measure of amount of substance. One mole is the formula weight of the substance expressed in grams. For example, for limonene, formula C10H16, the formula weight is (C = 12) (10 x 12) + (16 x 1) (H = 1) = 136 so that one mole of limonene is 136 grams of the compound. One mole of any substance contains the same number of units (atoms, molecules or ions). This is termed the Avogadro number, 6.022 x 1023 in scientific notation. 

When adding or subtracting numbers in scientific notation, the numbers must be converted to the same power of 10 as the measurement with the greatest power of 10. Once the numbers are all expressed to the same power of 10, the power of 10 is neither added nor subtracted in the calculation. 

The viscosity of carbon dioxide at 575.15 K has been measured as 0.0002682 p. In scientific notation we would write this as 2.682 p, realising that we need to include a factor of 0.0001 or 10 to correctly scale this. The value could thus be quoted as 2.682 x 10 " p. 

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. 

When measurements are multiplied or divided in scientific notation, their exponents are added or subtracted, respectively. 

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. 

The greater the number of digits in a measurement expressed in scientific notation, the more precise the measurement is. In this example, 2.00 x 10 L is the most precise data. 

Multiplying and Dividing Measurements Expressed in Scientific Notation... 

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