SI Base Units

SI Base Units

All measurements depend on the use of units that are well known and understood. The English system of measurement units (inches, feet, ounces, etc.) are not used in science because of the difficulty in converting from one unit to another. The metric system is used because all metric units are based on multiples of 10, making conversions very simple. The metric system was originally established in France in 1795. The International System of Units is a system of measurement based on the metric system. The acronym SI is commonly used to refer to this system and stands for the French term, Le Système International d’Unités . The SI was adopted by international agreement in 1960 and is composed of seven base units, as shown in Table below .

SI Base Units of Measurement

Quantity	SI Base Unit	Symbol
Length	meter	m
Mass	kilogram	kg
Temperature	kelvin	K
Time	second	s
Amount of a Substance	mole	mol
Electric Current	ampere	A
Luminous Intensity	candela	cd

Metric Prefixes

 

Metric Prefixes

Conversions between metric system units are straightforward because the system is based on powers of ten. For example, meters, centimeters, and millimeters are all metric units of length. There are 10 millimeters in 1 centimeter and 100 centimeters in 1 meter. Metric prefixes are used to distinguish between units of different size. These prefixes all derive from either Latin or Greek terms. For example, mega comes from the Greek word , meaning “great”

Table below lists the most common metric prefixes and their relationship to the central unit that has no prefix. Length is used as an example to demonstrate the relative size of each prefixed unit.

SI Prefixes

Prefix	Unit Abbrev.	Meaning	Example
giga	G	1,000,000,000	1 gigameter (Gm) = 10 9 m
mega	M	1,000,000	1 megameter (Mm) = 10 6 m
kilo	k	1000	1 kilometer (km) = 1000 m
hecto	h	100	1 hectometer (hm) = 100 m
deka	da	10	1 dekameter (dam) = 10 m
		1	1 meter (m)
deci	d	1/10	1 decimeter (dm) = 0.1 m
centi	c	1/100	1 centimeter (cm) = 0.01 m
milli	m	1/1000	1 millimeter (mm) = 0.001 m
micro	μ	1/1,000,000	1 micrometer (μm) = 10 -6 m
nano	n	1/1,000,000,000	1 nanometer (nm) = 10 -9 m
pico	p	1/1,000,000,000,000	1 picometer (pm) = 10 -12 m

Scientific Notation in Chemistry

 

Scientific Notation

Scientific notation is a way to express numbers as the product of two numbers: a coefficient and the number 10 raised to a power. It is a very useful tool for working with numbers that are either very large or very small.  As an example, the distance from Earth to the Sun is about 150,000,000,000 meters – a very large distance indeed.  In scientific notation, the distance is written as 1.5 × 10 ¹¹ m. The coefficient is the 1.5 and must be a number greater than or equal to 1 and less than 10.  The power of 10, or exponent, is 11 because you would have to multiply 1.5 by 10 ¹¹ to get the correct number.  Scientific notation is sometimes referred to as exponential notation. A summary of SI units is given in Table below .

SI Length and Volume Units

Length and Volume

Length is the measurement of the extent of something along its greatest dimension. The SI basic unit of length, or linear measure, is the meter (m). All measurements of length may be made in meters, though the prefixes listed in various tables will often be more convenient. The width of a room may be expressed as about 5 meters (m), whereas a large distance, such as the distance between New York City and Chicago, is better expressed as 1150 kilometers (km). Very small distances can be expressed in units such as the millimeter or the micrometer. The width of a typical human hair is about 20 micrometers (μm).

Volume is the amount of space occupied by a sample of matter. The volume of a regular object can be calculated by multiplying its length by its width by its height. Since each of those is a linear measurement, we say that units of volume are derived from units of length. The SI unit of volume is the cubic meter (m ³ ), which is the volume occupied by a cube that measures 1 m on each side. This very large volume is not very convenient for typical use in a chemistry laboratory. A liter (L) is the volume of a cube that measures 10 cm (1 dm) on each side. A liter is thus equal to both 1000 cm ³ (10 cm × 10 cm × 10 cm) and to 1 dm ³ . A smaller unit of volume that is commonly used is the milliliter (mL – note the capital L which is a standard practice). A milliliter is the volume of a cube that measures 1 cm on each side. Therefore, a milliliter is equal to a cubic centimeter (cm ³ ). There are 1000 mL in 1 L, which is the same as saying that there are 1000 cm ³ in 1 dm ³ .

 

 

 

SI Mass and Weight Units

 

Mass and Weight

Mass is a measure of the amount of matter that an object contains. The mass of an object is made in comparison to the standard mass of 1 kilogram. The kilogram was originally defined as the mass of 1 L of liquid water at 4°C (volume of a liquid changes slightly with temperature). In the laboratory, mass is measured with a balance ( Figure below ), which must be calibrated with a standard mass so that its measurements are accurate.

Other common units of mass are the gram and the milligram. A gram is 1/1000th of a kilogram, meaning that there are 1000 g in 1 kg. A milligram is 1/1000th of a gram, so there are 1000 mg in 1 g.

Mass is often confused with the term weight. Weight is a measure of force that is equal to the gravitational pull on an object. The weight of an object is dependent on its location. On the moon, the force due to gravity is about one sixth that of the gravitational force on Earth. Therefore, a given object will weigh six times more on Earth than it does on the moon. Since mass is dependent only on the amount of matter present in an object, mass does not change with location. Weight measurements are often made with a spring scale by reading the distance that a certain object pulls down and stretches a spring.

 

 

 

SI Kinetic Energy Units

 

SI Kinetic Energy Units

An object’s kinetic energy is the energy due to motion. Kinetic energy can be defined mathematically as

where

Energy is defined as the capacity to do work or to produce heat. As discussed previously, kinetic energy is one type of energy and is associated with motion. Another frequently encountered energy is potential energy , a type of energy that is stored in matter and released during a chemical reaction. The joule (J) is the SI unit of energy and is named after English physicist James Prescott Joule (1818-1889). If we go back to the equation for kinetic energy written above, we can put units in (kg for mass and m ² /s ² for velocity squared). Then, in terms of SI base units a joule is equal to a kilogram times meter squared divided by a second squared (kg • m ² /s ² ). Another common unit of energy that is often used is the calorie (cal), which is equivalent to 4.184 J.

 

 

 

Temperature and Temperature Scales

 

Temperature and Temperature Scales

Temperature is a measure of the average kinetic energy of the particles in matter. In everyday usage, temperature indicates a measure of how hot or cold an object is. Temperature is an important parameter in chemistry. When a substance changes from solid to liquid, it is because there was an increase in the temperature of the material. Chemical reactions usually proceed faster if the temperature is increased. Many unstable materials (such as enzymes) will be viable longer at lower temperatures.

Temperature Scales

The first thermometers were glass and contained alcohol, which expanded and contracted as the temperature changed. The German scientist, Daniel Gabriel Fahrenheit used mercury in the tube, an idea put forth by Ismael Boulliau. The Fahrenheit scale was first developed in 1724 and tinkered with for some time after that. The main problem with this scale is the arbitrary definitions of temperature. The freezing point of water was defined as 32°F and the boiling point as 212°F. The Fahrenheit scale is typically not used for scientific purposes.

The Celsius scale of the metric system is named after Swedish astronomer Anders Celsius (1701-1744). The Celsius scale sets the freezing point and boiling point of water at 0°C and 100°C respectively. The distance between those two points is divided into 100 equal intervals, each of which is one degree. Another term sometimes used for the Celsius scale is “centigrade” because there are 100 degrees between the freezing and boiling points of water on this scale. However, the preferred term is “Celsius.”

The Kelvin temperature scale is named after Scottish physicist and mathematician Lord Kelvin (1824-1907). It is based on molecular motion, with the temperature of 0 K, also known as absolute zero, being the point where all molecular motion ceases. The freezing point of water on the Kelvin scale is 273.15 K, while the boiling point is 373.15 K. Notice that here is no “degree” used in the temperature designation. Unlike the Fahrenheit and Celsius scales where temperatures are referred to as “degrees F” or “degrees C,” we simply designated temperatures in the Kelvin scale as kelvins.

As can be seen by the 100 kelvin difference between the two, a change of one degree on the Celsius scale is equivalent to the change of one kelvin on the Kelvin scale. Converting from the Kelvin scale to the Celsius scale or vice versa is easy, as you simply add or subtract 273.

 

To convert from Celsius to Farenheit, first subtract 32 from the temperature in Farenheit then divide by 1.8, in that order.

T_C = (T_F – 32)/1.8

 

 

 

Scientific Dimensional Analysis

Conversion Factors

Many quantities can be expressed in several different ways. The English system measurement of 4 cups is also equal to 2 pints, 1 quart, and ¼ of a gallon.

4 cups = 2 pints = 1 quart = 0.25 gallon

Notice that the numerical component of each quantity is different, while the actual amount of material that it represents is the same. That is because the units are different. We can establish the same set of equalities for the metric system:

1 meter = 10 decimeters = 100 centimeters = 100 millimeters

The metric system’s use of powers of 10 for all conversions makes this quite simple.

Whenever two quantities are equal, a ratio can be written that is numerically equal to 1. Using the metric examples above:

The is called a conversion factor . A conversion factor is a ratio of equivalent measurements. Because both 1 m and 100 cm represent the exact same length, the value of the conversion factor is 1. The conversion factor is read as “1 meter per 100 centimeters”. Other conversion factors from the cup measurement example can be:

Since the numerator and denominator represent equal quantities in each case, all are valid conversion factors.

Scientific Dimensional Analysis

Conversion factors are used in solving problems in which a certain measurement must be expressed with different units. When a given measurement is multiplied by an appropriate conversion factor, the numerical value changes, but the actual size of the quantity measured remains the same. Dimensional analysis is a technique that uses the units (dimensions) of the measurement in order to correctly solve problems. Dimensional analysis is best illustrated with an example.

Sample Problem: Dimensional Analysis

How many seconds are in a day?

Step 1: List the known quantities and plan the problem.

Known

• 1 day = 24 hours
• 1 hour = 60 minutes
• 1 minute = 60 seconds

Unknown

• 1 day = ? seconds

The known quantities above represent the conversion factors that we will use. The first conversion factor will have day in the denominator so that the “day” unit will cancel. The second conversion factor will then have hours in the denominator, while the third conversion factor will have minutes in the denominator. As a result, the unit of the last numerator will be seconds and that will be the units for the answer.

Step 2: Calculate

Applying the first conversion factor, the “d” unit cancels and 1 × 24 = 24. Applying the second conversion factor, the “h” unit cancels and 24 × 60 = 1440. Applying the third conversion factor, the “min” unit cancels and 1440 × 60 = 86,400. The unit that remains is “s” for seconds.

Step 3: Think about your result.

Seconds is a much smaller unit of time than a day, so it makes sense that there are a very large number of seconds in one day.

 

 

Metric Unit Conversions

 

Metric Unit Conversions

The metric system’s many prefixes allow quantities to be expressed in many different units. Dimensional analysis is useful to convert from one metric system unit to another.

Sample Problem: Metric Unit Conversions

A particular experiment requires 120 mL of a solution. The teacher knows that he will need to make enough solution for 40 experiments to be performed throughout the day. How many liters of solution should he prepare?

Step 1: List the known quantities and plan the problem.

Known

• 1 exp requires 120 mL
• 1 L = 1000 mL

Unknown

• L of solution for 40 exp

Since each experiment requires 120 ml of solution and the teacher needs to prepare enough for 40 experiments, multiply 120 by 40 to get 4800 mL of solution needed. Now you must convert ml to L by using a conversion factor.

Step 2: Calculate

Note that conversion factor is arranged so that the mL unit is in the denominator and thus cancels out, leaving L as the remaining unit in the answer.

Step 3: Think about your result.

A liter is much larger than a milliliter, so it makes sense that the number of liters required is less than the number of milliliters.

Two-Step Metric Unit Conversions

Some metric conversion problems are most easily solved by breaking them down into more than one step. When both the given unit and the desired unit have prefixes, one can first convert to the simple (unprefixed) unit, followed by a conversion to the desired unit. An example will illustrate this method.

Sample Problem 3.3: Two-Step Metric Conversion

Convert 4.3 cm to μm.

Step 1: List the known quantities and plan the problem.

Known

• 1 m = 100 cm
• 1 m = 10 ⁶ μm

Unknown

• 4.3 cm = ? μm

You may need to consult a table for the multiplication factor represented by each metric prefix. First convert cm to m, followed by a conversion of m to μm.

Step 2: Calculate

Each conversion factor is written so that unit of the denominator cancels with the unit of the numerator of the previous factor.

Step 3: Think about your result.

A micrometer is a smaller unit of length than a centimeter, so the answer in micrometers is larger than the number of centimeters given.

 

 

 

Derived Units

 

Dimensional Analysis and Derived Units

Some units are combinations of SI base units. A derived unit is a unit that results from a mathematical combination of SI base units. We have already discussed volume and energy as two examples of derived units.  Some others are listed in the Table below :

Density

 

 

Density

A golf ball and a table tennis ball are about the same size. However, the golf ball is much heavier than the table tennis ball. Now imagine a similar size ball made out of lead. That would be very heavy indeed! What are we comparing? By comparing the mass of an object relative to its size, we are studying a property called density. Density is the ratio of the mass of an object to its volume.

Density is an intensive property, meaning that it does not depend on the amount of material present in the sample. Water has a density of 1.0 g/mL. That density is the same whether you have a small glass of water or a swimming pool full of water. Density is a property that is constant for the particular identity of the matter being studied.

The SI units of density are kilograms per cubic meter (kg/m ³ ), since the kg and the m are the SI units for mass and length respectively. In everyday usage in a laboratory, this unit is awkwardly large. Most solids and liquids have densities that are conveniently expressed in grams per cubic centimeter (g/cm ³ ). Since a cubic centimeter is equal to a milliliter, density units can also be expressed as g/mL. Gases are much less dense than solids and liquids, so their densities are often reported in g/L. Densities of some common substances at 20°C are listed in the Table below.

Densities of Some Common Substances

Liquids and Solids	Density at 20°C (g/ml)	Gases	Density at 20°C (g/L)
Ethanol	0.79	Hydrogen	0.084
Ice (0°C)	0.917	Helium	0.166
Corn oil	0.922	Air	1.20
Water	0.998	Oxygen	1.33
Water (4°C)	1.000	Carbon dioxide	1.83
Corn syrup	1.36	Radon	9.23
Aluminum	2.70
Copper	8.92
Lead	11.35
Mercury	13.6
Gold	19.3

Since most materials expand as temperature increases, the density of a substance is temperature dependent and usually decreases as temperature increases.

You know that ice floats in water and it can be seen from the table that ice is less dense. Alternatively, corn syrup, being denser, would sink if placed into water.

Sample Problem: Density Calculations

An 18.2 g sample of zinc metal has a volume of 2.55 cm ³ . Calculate the density of zinc.

Step 1: List the known quantities and plan the problem.

Known

• mass = 18.2 g
• volume = 2.55 cm ³

Unknown

• density = ? g/cm ³

Use the equation for density, , to solve the problem.

Step 2: Calculate

Step 3: Think about your result.

If 1 cm ³ of zinc has a mass of about 7 grams, then 2 and a half cm ³ will have a mass about 2 and a half times as great. Metals are expected to have a density greater than that of water and zinc’s density falls within the range of the other metals listed above

Since density values are known for many substances, density can be used to determine an unknown mass or an unknown volume. Dimensional analysis will be used to ensure that units cancel appropriately.

Sample Problem: Using Density to Determine Mass and Volume

1. What is the mass of 2.49 cm ³ of aluminum?
2. What is the volume of 50.0 g of aluminum?

Step 1: List the known quantities and plan the problem.

Known

• density = 2.70 g/cm ³
• 1. volume = 2.49 cm ³
• 2. mass = 50.0 g

Unknown

• 1. mass = ? g
• 2. volume = ? cm ³

Use the equation for density, , and dimensional analysis to solve each problem.

Step 2: Calculate

1. 
2. 

In problem 1, the mass is equal to the density multiplied by the volume. In problem 2, the volume is equal to the mass divided by the density.

Step 3: Think about your results.

Because a mass of 1 cm ³ of aluminum is 2.70 g, the mass of about 2.5 cm ³ should be about 2.5 times larger. The 50 g of aluminum is substantially more than its density, so that amount should occupy a relatively large volume.

 

 

 

Accuracy and Precision

 

Accuracy and Precision

In everyday speech, the terms accuracy and precision are frequently used interchangeably.  However, their scientific meanings are quite different.  Accuracy is a measure of how close a measurement is to the correct or accepted value of the quantity being measured.  Precision is a measure of how close a series of measurements are to one another.  Precise measurements are highly reproducible, even if the measurements are not near the correct value.

Darts thrown at a dartboard are helpful in illustrating accuracy and precision

Assume that three darts are thrown at the dartboard, with the bulls-eye representing the true, or accepted, value of what is being measured.  A dart that hits the bulls-eye is highly accurate, whereas a dart that lands far away from the bulls-eye displays poor accuracy.  The Figure above demonstrates four possible outcomes.

1. The darts have landed far from each other and far from the bulls-eye.  This grouping demonstrates measurements that are neither accurate, nor precise.
2. The darts are close to one another, but far from the bulls-eye.  This grouping demonstrates measurements that are precise, but not accurate.  In a laboratory situation, high precision with low accuracy often results from a systematic error.  Either the measurer makes the same mistake repeatedly or the measuring tool is somehow flawed.  A poorly calibrated balance may give the same mass reading every time, but it will be far from the true mass of the object.
3. The darts are not grouped very near to each other, but are generally centered around the bulls-eye.  This demonstrates poor precision, but fairly high accuracy.  This situation is not desirable in a lab situation because the “high” accuracy may simply be random chance and not a true indicator of good measuring skill.
4. The darts are grouped together and have hit the bulls-eye.  This demonstrates high precision and high accuracy.  Scientists always strive to maximize both in their measurements.

 

 

Percent Error

 

Percent Error

An individual measurement may be accurate or inaccurate, depending on how close it is to the true value.  Suppose that you are doing an experiment to determine the density of a sample of aluminum metal.  The accepted value of a measurement is the true or correct value based on general agreement with a reliable reference.  For aluminum the accepted density is 2.70 g/cm ³ .  The experimental value of a measurement is the value that is measured during the experiment.  Suppose that in your experiment you determine an experimental value for the aluminum density to be 2.42 g/cm ³ .  The error of an experiment is the difference between the experimental and accepted values.

If the experimental value is less than the accepted value, the error is negative.  If the experimental value is larger than the accepted value, the error is positive.  Often, error is reported as the absolute value of the difference in order to avoid the confusion of a negative error.  The percent error is the absolute value of the error divided by the accepted value and multiplied by 100%.

To calculate the percent error for the aluminum density measurement, we can substitute the given values of 2.45 g/cm ³ for the experimental value and 2.70 g/cm ³ for the accepted value.

If the experimental value is equal to the accepted value, the percent error is equal to 0.  As the accuracy of a measurement decreases, the percent error of that measurement rises.

 

 

 

Measurement Uncertainty

 

Measurement Uncertainty

Some error or uncertainty always exists in any measurement. The amount of uncertainty depends both upon the skill of the measurer and upon the quality of the measuring tool. While some balances are capable of measuring masses only to the nearest 0.1 g, other highly sensitive balances are capable of measuring to the nearest 0.001 g or even better. Many measuring tools such as rulers and graduated cylinders have small lines which need to be carefully read in order to make a measurement. The figure below shows two rulers making the same measurement of an object (indicated by the blue arrow).

With either ruler, it is clear that the length of the object is between 2 and 3 cm. The bottom ruler contains no millimeter markings. With that ruler, the tenths digit can be estimated and the length may be reported as 2.5 cm.  However, another person may judge that the measurement is 2.4 cm or perhaps 2.6 cm. While the 2 is known for certain, the value of the tenths digit is uncertain.

The top ruler contains marks for tenths of a centimeter (millimeters). Now the same object may be measured as 2.55 cm. The measurer is capable of estimating the hundredths digit because he can be certain that the tenths digit is a 5. Again, another measurer may report the length to be 2.54 cm or 2.56 cm. In this case, there are two certain digits (the 2 and the 5), with the hundredths digit being uncertain. Clearly, the top ruler is a superior ruler for measuring lengths as precisely as possible.

 

 

 

Glossary

Significant Figures

Significant Figure

 

The significant figures in a measurement consist of all the certain digits in that measurement plus one uncertain or estimated digit. In the ruler illustration below, the bottom ruler gave a length with 2 significant figures, while the top ruler gave a length with 3 significant figures. In a correctly reported measurement, the final digit is significant but not certain. Insignificant digits are not reported. With either ruler, it would not be possible to report the length as 2.553 cm as there is no possible way that the thousandths digit could be estimated. The 3 is not significant and would not be reported.

When you look at a reported measurement, it is necessary to be able to count the number of significant figures.  The Table below details the rules for determining the number of significant figures in a reported measurement. For the examples in the table, assume that the quantities are correctly reported values of a measured quantity.

Rounding

 

Rounding

Before dealing with the specifics of the rules for determining the significant figures in a calculated result, we need to be able to round numbers correctly. To round a number, first decide how many significant figures the number should have. Once you know that, round to that many digits, starting from the left. If the number immediately to the right of the last significant digit is less than 5, it is dropped and the value of the last significant digit remains the same. If the number immediately to the right of the last significant digit is greater than or equal to 5, the last significant digit is increased by 1.

Consider the measurement 207.518 m. Right now, the measurement contains six significant figures. How would we successively round it to fewer and fewer significant figures? Follow the process as outlined in Table below .

Uncertainty in Multiplication and Division

 

Uncertainty in Multiplication and Division

The density of a certain object is calculated by dividing the mass by the volume.  Suppose that a mass of 37.46 g is divided by a volume of 12.7 cm ³ .  The result on a calculator would be:

The value of the mass measurement has four significant figures, while the value of the volume measurement has only three significant figures.  For multiplication and division problems, the answer should be rounded to the same number of significant figures as the measurement with the least number of significant figures.  Applying this rule results in a density of 2.95 g/cm ³ , for three significant figures – the same as the volume measurement.

Sample Problem:  Significant Figures in Calculations

Perform the following calculations, rounding the answers to the appropriate number of significant figures.

1. 
2. 

Step 1:  Plan the problem.

Analyze each of the measured values to determine how many significant figures should be in the result.  Perform the calculation and round appropriately.  Apply the correct units to the answer. When multiplying or dividing, the units are also multiplied or divided.

Step 2:  Calculate

1.   Round to two significant figures because 0.048 has two.
2.  Round to three significant figures because 5.81 has three.

 

 

 

Uncertainty in Addition and Subtraction

 

 

Uncertainty in Addition and Subtraction

Consider two separate mass measurements: 16.7 g and 5.24 g. The first mass measurement (16.7 g) is known only to the tenths place or to one digit after the decimal point. There is no information about its hundredths place and so that digit cannot be assumed to be zero. The second measurement (5.24 g) is known to the hundredths place or to two digits after the decimal point.

When these masses are added together, the result on a calculator is 16.7 + 5.24 = 21.94 g. Reporting the answer as 21.94 g suggests that the sum is known all the way to the hundredths place. However that cannot be true because the hundredths place of the first mass was completely unknown. The calculated answer needs to be rounded in such a way as to reflect the certainty of each of the measured values that contributed to it. For addition and subtraction problems, the answer should be rounded to the same number of decimal places as the measurement with the least number of decimal places. The sum of the above masses would be properly rounded to a result of 21.9 g.

When working with whole numbers, pay attention to the last significant digit that is to the left of the decimal point and round your answer to that same point. For example, consider the subtraction: 78,500 m – 362 m. The calculated result is 78,138 m. However, the first measurement is known only to the hundreds place, as the 5 is the last significant digit. Rounding the result to that same point means that the correct result is 78,100 m.