Numbers and Significant Figures

We often encounter very small and very large numbers in chemistry problems. For example, pesticide production in the world exceeds millions of tons, whereas pesticide residues that may harm animals or humans can have masses as small as nanograms. For either type of number, scientific notation is useful. Numbers written using scientific notation factor out all powers of ten and write them separately. Thus the number 54,000 is written as 5.4 x lO. This notation is equivalent to 5.4 X 10,000, which clearly is 54,000. Small numbers can also be written in scientific notation using negative powers of ten, because 10 is identical to 1/10. The number 0.000042 is 4.2 x 10 in scientific notation. 

For example, an almanac lists the population of Canada as 31,110,600. Suppose a study concluded that 24% of the people who live in Canada speak French. Based on this information alone, the wager is that 7,466,544 Canadians speak 

Scientific notation provides an advantage here, because any digits shown in the number are significant. There are no zeros needed for placing the decimal. 

An alloy contains 2.05% of some impurity. How many significant figures are reported in this value  

Use the general rule for significant figures digits reported are significant unless they are zeros whose sole purpose is to position the decimal place. 

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Measured Numbers and Significant Figures LEARNING GOAL Identify a number as measured or exact determine the number of significant figures in a measured number. 

Exact numbers, such as the stoichiometric coefficients in a chemical formula or reaction, and unit conversion factors, have an infinite number of significant figures. A mole of CaCb, for example, contains exactly two moles of chloride and one mole of calcium. In the equality... 

Calculate the molar concentration of NaCl, to the correct number of significant figures, if 1.917 g of NaCl is placed in a beaker and dissolved in 50 mF of water measured with a graduated cylinder. This solution is quantitatively transferred to a 250-mF volumetric flask and diluted to volume. Calculate the concentration of this second solution to the correct number of significant figures. 

Note that the values given for Wq limit the number of significant figures to four for H2 and to three for F2 and I2. 

Much of the additional material is taken up by what 1 have called Worked examples . These are sample problems, which are mostly calculations, with answers given in some detail. There are seventeen of them scattered throughout the book in positions in the text appropriate to the theory which is required. 1 believe that these will be very useful in demonstrating to the reader how problems should be tackled. In the calculations, 1 have paid particular attention to the number of significant figures retained and to the correct use of units. 1 have stressed the importance of putting in the units in a calculation. In a typical example, for the calculation of the rotational constant B for a diatomic molecule from the equation... 

In each example, the initial values of the factors are expressed in U.S. customary units, and the dimensionless value is calculated. Then the factors are converted to SI units, and the dimensionless value is recalculated. The two dimensionless values will be approximately the same. (Small variations occur due to the number of significant figures carried in the solution.)... 

Significant figures provide an indication of the precision with which a quantity is measured or known. The last digit represents, in a quantitative sense, some degree of doubt. For example, a measurement of 8.12 inches implies tliat Uie actual quantity is somewhere between 8.315 and 8.325 inches. This applies to calculated and measured quantihes quantities tliat are known exactly (e.g., pure integers) have an infinite number of significant figures. 

Count the number of significant figures in the numerator and in the denominator the smaller of these two numbers is the number of significant figures in the quotient... 

The numbers 1.8 and 32 are exact Hence they do not limit the number of significant figures in a temperature conversion that limit is determined only by the precision of the thermometer used to measure temperature. 

The abbreviation sf denotes the number of significant figures in the data. The frequencies, wavelengths, and energies are typical values they should not be regarded as precise. 

The digits in a reported measurement are called the significant figures. There are two significant figures (written 2 sf) in 1.2 cm3 and 3 sf in 1.78 g. Section A describes how to find the number of significant figures in a measurement. 

E (2) = +0.77 V, potentials should be identical but differ because of the limited number of significant figures used to derive and evaluate the equations. 

In a sequence of computations, adjusting the number of significant figures in intermediate results can lead to errors in the final value. Instead, wait until the computations are complete, and then express the final value with the appropriate number of significant figures. 

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