﻿ MEASUREMENTS AND CALCULATIONS - Introduction to Chemistry - REVIEW OF MAJOR TOPICS - SAT Subject Test Chemistry ﻿

## PART 2 ## REVIEW OF MAJOR TOPICS ### MEASUREMENTS AND CALCULATIONS

The student of chemistry must be able to make good observations. Observations are either qualitative or quantitative. Qualitative observations involve descriptions of the nature of the substances under investigation. Quantitative observations involve making measurements to describe the substances under observation. The chemistry student must also be able to use correct measurement terms and the required mathematical skills to solve the problems. The following sections review these topics.

Metric System

It is important that scientists around the world use the same units when communicating information. For this reason, scientists use the modernized metric system, designated in 1960 by the General Conference on Weights and Measures as the International System of Units. This is commonly known as SI, an abbreviation for the French name Le Système International d’Unités. It is now the most common system of measurement in the world. There are minor differences between the SI and metric systems. For the most part, the quantities are interchangeable.

TIP Only metric units are used on the SAT test.

The reason SI is so widely accepted is twofold. First, it uses the decimal system as its base. Second, many units for various quantities are defined in terms of units for simpler quantities.

There are seven basic units that can be used to express the fundamental properties of measurement. These are called the SI base units and are shown in the table that follows. * The candela is rarely used in chemistry.

Other SI units are derived by combining prefixes with a base unit. The prefixes represent multiples or fractions of 10. The following table gives some basic prefixes used in the metric system. For an example of how a prefix works in conjunction with the base word, consider the term kilometer. The prefix kilo- means “multiply the root word by 1,000,” so a kilometer is 1,000 meters. By the same reasoning, a millimeter is 1/1,000 meter.

Because of the prefix system, all units and quantities can be easily related by some factor of 10. Here is a brief table of some metric unit equivalents.

TIP

Know these relationships. Length 10 millimeters (mm) = 1 centimeter (cm) 100 cm = 1 meter (m) 1,000 m = 1 kilometer (km)
 Volume 1,000 milliliters (mL) = 1 liter (L) 1,000 cubic centimeters (cm3) = 1 liter 1 mL = 1 cm3
 Mass 1,000 milligrams (mg) = 1 gram (g) 1,000 g = 1 kilogram (kg)

A unit of length, used especially in expressing the length of light waves, is the nanometer, abbreviated as nm and equal to 10−9 meter.

Because in the United States measurements are occasionally reported in units of the English system, it is important to be aware of some metric to English system equivalents. Some common conversion factors are shown in the following table.

 2.54 centimeters = 1 inch 1 meter = 39.37 inches (10% longer than 1 yard) 28.35 grams = 1 ounce 454 grams = 1 pound 1 kilogram = 2.2 pounds 0.946 liter = 1 quart 1 liter (5% larger than 1 quart) = 1.06 quarts

TIP For your information only; metric is used on the test.

The metric system standards were chosen as natural standards. The meter was first described as the distance marked off on a platinum-iridium bar but now is defined as the length of the path traveled by light in a vacuum during a time interval of 1/2.99792458 × 108 second.

There are some interesting relationships between volume and mass units in the metric system. Because water is most dense at 4°C, the gram was intended to be 1 cubic centimeter of water at this temperature. This means, then, that:

1,000 cm3 = 1 L of water @ 4°C

1,000 cm3 of water weighs 1,000 g @ 4°C.

Therefore

1 L of water @ 4°C weighs 1 kg,

and

1 mL of water @ 4°C weighs 1 g. When 1 L is filled with water @ 4°C, it has a mass of 1 kg.

Temperature Measurements

The most commonly used temperature scale in scientific work is the Celsius scale. It gets its name from the Swedish astronomer Anders Celsius and dates back to 1742. For a long time it was called the centigrade scale because it is based on the concept of dividing the distance on a thermometer between the freezing point of water and its boiling point into 100 equal markings or degrees.

Another scale is based on the lowest theoretical temperature (called absolute zero). This temperature has never actually been reached, but scientists in laboratories have reached temperatures within about a thousandth of a degree above absolute zero. Lord William Kelvin proposed this scale, on which a degree is the same size as a Celsius degree and which is referred to as the Kelvin scale. Through experiments and calculations, it has been determined that absolute zero is 273.15 degrees below zero on the Celsius scale. This figure is usually rounded off to –273°C.

The diagram and conversion formulas that follow give the graphic and algebraic relationships among three temperature scales: the Celsius and Kelvin, commonly used in chemistry, and the Fahrenheit.

TIP Kelvin and °C units are used on the SAT test.  Note: In Kelvin notation, the degree sign is omitted: 283 K. The unit is the kelvin, abbreviated as K.

TIP REMEMBER The Kelvin unit is designated with K and does not have the “°” symbol.

EXAMPLE 1: EXAMPLE 2: Heat Measurements

Heat energy (or just heat) is a form of energy that transfers among particles in a substance (or system) by means of the kinetic energy of those particles. In other words, under kinetic theory, heat is transferred by particles bouncing into each other.

The scales above are used to measure the degree of heat. A pail and a thimble can both be filled with water at 100° Celsius. The water in both measures the same degree of heat. However, the pail of water has a greater quantity of heat. This could be easily demonstrated by the amount of ice that could be melted by the water in these two containers. Obviously, the pail of water at 100° Celsius will melt more ice than will a thimble full of water at the same temperature. Therefore, the pail of water contains a greater number of calories of heat. The calorie unit is used to measure the quantity of heat. It is defined as the amount of heat needed to raise the temperature of 1 gram of water by 1 degree on the Celsius scale. This is a rather small unit to measure the quantities of heat involved in most chemical reactions. Therefore, the kilocalorie is more often used. The kilocalorie equals 1,000 calories. It is the quantity of heat that will increase the temperature of 1 kilogram of water by 1 degree on the Celsius scale. Although the calorie is commonly used in everyday usage with regard to food, the SI unit for heat energy is the joule. It is abbreviated as J and, because it is a rather small unit, it is commonly given in kilojoules (kJ). The relationship between the calorie and the joule is that 1 calorie equals 4.18 joules.

Problems involving heat transfers in water are called water calorimetry problems and are explained in “Water Calorimetry Problems” Chatper 7.

TIP Because the joule is rather small, kJ is used most often.

Exponential Notation

When students must do mathematical operations with numerical figures, the exponential notation system is very useful. Basically this system uses the exponential means of expressing figures. With large numbers, such as 3,630,000., move the decimal point to the left until only one digit remains to the left (3.630000) and then indicate the number of moves of the decimal point as the exponent of 10 (3.63 × 106). With a very small number such as 0.000000123, move the decimal point to the right until only one digit is to the left (0000001.23) and then express the number of moves as the negative exponent of 10 (1.23 × 10−7).

TIP Scientific notation is based on exponents of 10.

With numbers expressed in this exponential form, you can now use your knowledge of exponents in mathematical operations. An important fact to remember is that in multiplication you add the exponents of 10, and in division you subtract the exponents. Addition and subtraction of two numbers expressed in scientific notation can be performed only if the numbers have the same exponent.

TIP REMEMBER Only one digit can be to the left of the decimal point.

EXAMPLES:

Multiplications:

(2.3 × 105)(5.0 × 10−12). Multiplying the first numbers, you get 11.5, and addition of the exponents gives 10−7. Now, changing to a number with only one digit to the left of the decimal point gives you 1.15 × 10−6 for the answer.

Try these:

(5.1 × 10−6)(2 × 10−3) = 10.2 × 10−9 = 1.02 × 10−8
(3 × 105)(6 × 103) = 18 × 108 = 1.8 × 109

Divisions:

(1.5 × 103) ÷ (5.0 × 10−2) = 0.3 × 105 = 3 × 104
(2.1 × 10−2) ÷ (7.0 × 10−3) = 0.3 × 101 = 3
(Notice that in division the exponents of 10 are subtracted.)

Addition and subtraction:

(4.2 × 104 kg) + (7.9 × 103 kg) = (4.2 × 104 kg) + (0.79 × 104 kg) (note that the exponents of 10 are now the same) = 4.99 × 104 kg This can be rounded to 5.0 × 104 kg.

(6.02 × 10−3) – (2.41 × 10−4) = (6.02 × 10−3) – (.241 × 10−3) (note that the exponents of 10 are now the same) = 5.779 × 10−3 or 5.8 × 10−3 when rounded to two significant figures.

Dimensional Analysis (Factor-Label Method of Conversion)

When you are working problems that involve numbers with units of measurement, it is convenient to use this method so that you do not become confused in the operations of multiplication or division. For example, if you are changing 0.001 kilogram to milligrams, you set up each conversion as a fraction so that all the units will factor out except the one you want in the answer. TIP Cancel out all units except the one for the answer.

Notice that the kilogram is made the denominator in the first fraction to be factored with the original kilogram unit. The numerator is equal to the denominator except that the numerator is expressed in smaller units. The second fraction has the gram unit in the denominator to be factored with the gram unit in the preceding fraction. The answer is in milligrams because this is the only unit remaining and it assures you that the correct operations have been performed in the conversion.

ANOTHER EXAMPLE: The factor-label method is used in examples throughout this book.

Precision, Accuracy, and Uncertainty

Two other factors to consider in measurement are precision and accuracy. Precision indicates the reliability or reproducibility of a measurement. Accuracy indicates how close a measurement is to its known or accepted value.

TIP Accuracy is how close you have come to the true value.

For example, suppose you were taking a reading of the boiling point of pure water at sea level. Using the same thermometer in three trials, you record 96.8, 96.9, and 97.0 degrees Celsius. Since these figures show a high reproducibility, you can say that they are precise. However, the values are considerably off from the accepted value of 100 degrees Celsius, so we say they are not accurate. In this example we probably would suspect that the inaccuracy was the fault of the thermometer.

TIP Precision is how repeatable the results are.

Regardless of precision and accuracy, all measurements have a degree of uncertainty. This is usually dependent on one or both of two factors—the limitation of the measuring instrument and the skill of the person making the measurement. Uncertainty can best be shown by example. The graduated cylinder in the illustration contains a quantity of water to be measured. It is obvious that the quantity is betwen 30 and 40 milliliters because the meniscus lies between these two marked quantities. Now, checking to see where the bottom of the meniscus lies with reference to the ten intervening subdivisions, we see that it is between the fourth and fifth. This means that the volume lies between 34 and 35 milliliters. The next step introduces the uncertainty. We have to guess how far the reading is between these two markings. We can make an approximate guess, or estimate, that the level is more than 0.2 but less than 0.4 of the distance. We therefore report the volume as 34.3 milliliters. The last digit in any measurement is an estimate of this kind and is uncertain.

Significant Figures

Any time a measurement is recorded, it includes all the digits that are certain plus one uncertain digit. These certain digits plus the one uncertain digit are referred to as significant figures. The more digits you are able to record in a measurement, the less relative uncertainty there is in the measurement. The following table summarizes the rules of significant figures. One last rule deals with final zeros in a whole number. These zeros may or may not be significant, depending on the measuring instrument. For instance, if an instrument that measures to the nearest mile is used, the number 3,000 miles has four significant figures. If, however, the instrument in question records miles to the nearest thousands, there is only one significant figure. The number of significant figures in 3,000 could be one, two, three, or four, depending on the limitation of the measuring device.

This problem can be avoided by using the system of scientific notation. For this example, the following notations would indicate the numbers of significant figures:

 3 × 103 one significant figure 3.0 × 103 two significant figures 3.00 × 103 three significant figures 3.000 × 103 four significant figures

Calculations with Significant Figures

When you do calculations involving numbers that do not have the same number of significant figures in each, keep the following two rules in mind.

TIP RULE Your answer cannot have more significant figures than the quantity having the fewest number of significant figures.

First, in multiplication and division, the number of significant figures in a product or a quotient of measured quantities is the same as the number of significant figures in the quantity having the smaller number of significant figures.

EXAMPLE 1

 Problem Unrounded answer Answer rounded to thecorrect number ofsignificant figures 4.29 cm × 3.24 cm = 13.8996 cm2 = 13.9 cm2

Explanation: Both measured quantities have three significant figures. Therefore, the answer should be rounded to three significant figures.

EXAMPLE 2

 Problem Unrounded answer Answer rounded to thecorrect number ofsignificant figures 4.29 cm × 3.2 cm = 13.728 cm2 = 14 cm2

Explanation: One of the measured quantities has only two significant figures. Therefore, the answer should be rounded to two significant figures.

EXAMPLE 3

 Problem Unrounded answer Answer rounded to thecorrect number ofsignificant figures 8.47 cm2/4.26 cm = 1.9882629 cm = 1.99 cm

Explanation: Both measured quantities have three significant figures. Therefore, the answer should be rounded to three significant figures.

Second, when adding or subtracting measured quantities, the sum or difference should be rounded to the same number of decimal places as the quantity having the least number of decimal places.

EXAMPLE 1

 Problem Unrounded answer Answer rounded to thecorrect number ofsignificant figures 3.56 cm 2.6 cm +6.12 cm Total= 12.28 cm 12.3 cm

Explanation: One of the quantities added has only one decimal place. Therefore, the answer should be rounded to only one decimal place.

EXAMPLE 2

 Problem Unrounded answer Answer rounded to thecorrect number ofsignificant figures 3.514 cm –2.13 cm Difference= 1.384 cm= 1.38 cm

Explanation: One of the quantities has only two decimal places. Therefore, the answer should be rounded to only two decimal places.

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