Über die Autorin bzw. den Autor

Charles Henrickson, Ph.D., is a retired professor of Chemistry at Western Kentucky University.

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CliffsNotes Chemistry Practice Pack

John Wiley & Sons

Chapter One

At the core of any science is measurement. Being able to measure volumes, pressures, masses, and temperatures as well as the ability to count atoms and molecules allows chemists to understand nature more precisely. Modern science uses the International System of Units (SI) that was adopted worldwide in 1960. The metric system of measurement, which is consistent with the International System, is widely used in chemistry and is the principal system used in this book.

Chemists often have to work with numbers that are very, very small or very, very large. It is more convenient to express numbers of this kind in scientific notation, so that is the first topic to look at in this chapter.

Writing Numbers in Scientific Notation

It is likely that you have already seen numbers expressed in scientific notation on your calculator. With only 8 or 9 spaces to display numbers, calculators must resort to scientific notation to show very small or very large numbers. In scientific notation a number is expressed in this form

a x [10.sup.p]

where "a" is a number between 1 and 10 (often a decimal number) and "p" is a positive or negative whole number written as an exponent on 10, often called the power of 10. The average distance from the earth to the sun is 93,000,000 miles, a very large number. In scientific notation this would be 9.3 x [10.sup.7] miles. The power of 7 equals the number of places a decimal point would be moved from the right end of 93,000,000 to the left to get an "a" value between 1 and 10 (9.3). The [10.sup.7] term equals 10,000,000 (10 x 10 x 10 x 10 x 10 x 10 x 10) and when multiplied by 9.3 would restate the original number in conventional form.

9.3 x [10.sup.7] miles = 9.3 x 10,000,000 miles = 93,000,000 miles

Likewise, the year 1492 would be written 1.492 x [10.sup.3] in scientific notation.

Small numbers, those less than 1, are handled in a similar way, except the decimal point has to be moved to the right to get an "a" value between 1 and 10, and the "p" exponent is a negative number. For example, an atom of gold has a diameter of 0.000000342 meter. The decimal must be moved 7 places to the right to get 3.42, and the number is stated as 3.42 x [10.sup.-7] meter in scientific notation. The [10.sup.-7] term equals 1 divided by 10,000,000.

3.42 x [10.sup.-7] meter = 3.42 x 1/10,000,000 meter = 0.000000342 meter

Similarly, the number 0.000045 would be written 4.5 x [10.sup.5].

One last thing: If you are given a number between 1 and 10 and need to write it in scientific notation, the power on 10 would be zero. In scientific notation, the number 8 would be written 8 x [10.sup.0]. In mathematics, [10.sup.0] equals 1.

Example Problems

1. Express these numbers in scientific notation.

(a) 22,500,000

Answer: 2.25 x [10.sup.7]

This is a large number, so the decimal is moved 7 places to the left to form 2.25, and 7 is written as a positive exponent on 10.

(b) 0.0006

Answer: 6 x [10.sup.4]

This is a small number, so the decimal must be moved 4 places to the right to form 6, and 4 is written as a negative exponent on 10.

(c) 602,200,000,000,000,000,000,000

Answer: 6.022 x [10.sup.23]

This is a very large number, so the decimal is moved 23 places to the left to form 6.022, and 23 is written as a positive exponent on 10.

2. Express these numbers in conventional form.

(a) 6.35 x [10.sup.5]

Answer: 635,000

The power of 10 is 5, a positive number, so the decimal point is moved 5 places to the right. (b) 2.4 x [10.sup.-3]

Answer: 0.0024

The power of 10 is -3, a negative number, so the decimal point is moved 3 places to the left.

Work Problems

1. Express these numbers in scientific notation.

(a) 1945 (b) 0.00000255 (c) 388000000000 (d) 0.023

2. Express these numbers in conventional form: (a) 7.55 x [10.sup.-4] (b) 8.80 x [10.sup.2]

Worked Solutions

1. (a) 1.945 x [10.sup.3]. Because this is a large number, the decimal is moved 3 places to the left to form 1.945. The exponent on 10 is 3.

(b) 2.55 x [10.sup.-6]. This is a small number, so the decimal is moved 6 places to the right to form 2.55. The exponent on 10 is -6.

(c) 3.88 x [10.sup.11]. This is a large number, so the decimal is moved 11 places to the left to form 3.88. The exponent on 10 is 11.

(d) 2.3 x [10.sup.-2]. Because this is a small number, the decimal is moved 2 places to the right to form 2.3. The exponent on 10 is -2.

2. (a) 0.000755. The power of 10 is -4, a negative number, so the decimal point is moved 4 places to the left.

(b) 880. The power of 10 is 2, a positive number, so the decimal point is moved 2 places to the right.

Significant Figures and Rounding Off Numbers

It is not possible to measure anything exactly; there will always be some amount of uncertainty. In many cases, the tool used to do the measurement causes the uncertainty. An inexpensive laboratory balance, for example, measures the mass of a gold ring to be 2.83 grams, while a more expensive analytical balance measures the mass to a greater accuracy, 2.8275 grams. The greater accuracy of the analytical balance is reflected in the larger number of digits in the numerical value of the mass. In either number, 2.83 or 2.8275, the right-most digit is the only digit that is not known with certainty. The mass of the ring is closer to 2.83 grams than to 2.82 or 2.84 grams on the first balance (2.83 0.01), and closer to 2.8275 grams than to 2.8274 or 2.8276 grams on the second (2.8275 0.0001). In both cases, all the digits are certain except the last one. The number of digits shown in a measured value (the certain digits + the one uncertain digit) indicates the accuracy of that value. These digits are referred to as significant digits or, more commonly, significant figures (sig. figs.).

Counting Significant Figures

You need to know how to count the number of significant figures in a number, because they affect the way answers are stated in calculations. Zeros can be a problem. A zero may or may not be significant depending on how it is used. To handle this "zero problem," follow this set of six rules:

Rule 1. All nonzero digits (1, 2, 3, 4, 5, 6, 7, 8, and 9) are always significant and must be counted.

Rule 2. A zero standing alone to the left of a decimal point is not significant. For example, in 0.63 and 0.0055, the 0 to the left of the decimal only helps you see the decimal point. It has no other use.

Rule 3. For a number less than 1, any zeros between the decimal point and the first nonzero digit are not significant. These zeros are simply placing the decimal point. The zeros in bold type in 0.00457 and 0.0000864 are not significant. Both numbers have three significant figures: 457 in the first number and 864 in the second.

Rule 4. A zero between two nonzero digits is significant. In 2.0056 and 0.0040558, both numbers have 5 significant figures. Because the second number is less than 1, only the 4, 0,...