Brick 002 - Prior Knowledge - Scientific Measurement and Units

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Prior Knowledge What You Should Already Know – Measurement and Units LE 05/15

Brick 002: NAME: __________________________________

What You Should Already Know – Measurement and Units

Dimensional Analysis and Data Processing

Sophomore Topics B and C Block System: 005-026

Dimensional analysis and unit conversions

In introductory chemistry, you should have learned a simple method for converting units, commonly known

as dimensional analysis. This method employed multiplying measured quantities by unity fractions in

order to cancel the starting units, replacing them with desired units. Hopefully, you can recall the following:

Let’s say I wanted to convert 3 hours into minutes. First, write out the problem.

3 ℎ = ?

Next, write out what the problem gives you, and put it over the number 1.

3 ℎ

1

Following that, write a fraction line next to it, and think about what units to cancel. I want to get rid of hours

and replace it with minutes. I know that there are 60 minutes in one hour, so I can write that. I put hours on

the bottom of the second fraction, and minutes on the top. This has the effect of canceling the units I don’twant, and replacing them with the units that I do want. Remember, the numerator of the second fraction

(60 minutes) and the denominator (1 hour) must be equal for this procedure to work.

3 ℎ

1 ×

60

1 ℎ = ???

If I stop there, I will have created an arrangement of fractions that, when combined together, will convert 3

hours into minutes. The units have cancelled, so all that is left is to plug numbers into a calculator and

perform a mathematical operation to get a numerical answer. Rather than combining numbers and dealing

with the rules of fraction math, or using the parenthesis function on your calculator to keep numerator and

denominator separate, it is easier to simply multiply the numbers on top and divide the numbers on thebottom.

3 ℎ

1 ×

60

1 ℎ = 3 × 60 = 180





This method works whether the units in question are getting bigger or smaller:

Here is the calculation of how many feet are in 6 inches, requiring the division function.

6 ℎ

1 ×

1

12 ℎ = 6 ÷ 12 = 0.5

12 inches ends up in the denominator. Therefore it is divided in order to get the proper answer of 0.5 feet.

Also, the same method works for multiple-step conversions:

Okay, now let’s say I wanted to convert 4 hours into seconds. This is a slightly more complex problem, but it

begins the same way. To start out, write out what the problem gives you, and put it over the number 1.

4 ℎ

1

Write a fraction line next to it, and think about what units to cancel.

4 ℎ

1 ×

? ? ?

1 ℎ

I want to get rid of hours and replace it with seconds, but I don’t know how many seconds are in one hour. I

do know that there are 60 minutes in one hour, so I can do that, canceling the units as I go.

4 ℎ

1 ×60

1 ℎ

If I stop there, I will have converted 4 hours into minutes. A quick glance back at the problem tells me that I

have to keep going until I have seconds as my units. So, I continue on, canceling minutes with seconds, just

like I canceled hours earlier.

4 ℎ

1 ×

60

1 ℎ ×

60

1

Finally, multiply all the fractions together, and get the answer.

4 ℎ1

× 60 1 ℎ

× 60 1

= 4 × 60 × 60 = 14400

Again, this method works because the units are properly canceled. Each of the conversion fractions written

has a numerator that is mathematically equal to its denominator. 60 minutes is the same as 1 hour. 60

seconds is the same as 1 minute.





The method works for squared and cubed units:

However, what happens if we want to solve the following problem?

1

= ? ℎ

What is a square foot? What are square inches? These are units commonly associated with the

measurement of area. Area can measured my multiplying the length of a region by its width. The units are

also multiplied, and can be rewritten in the following way for clarity.

1 = 1 ∙

This now presents itself as a two-step conversion problem. In order to convert square feet to square inches,

feet must be converted to inches twice.

1

1 =

1 ∙

1 ×

12 ℎ

1 ×

12 ℎ

1 = 1 × 12 × 12 = 144 ℎ

And it works for compound units:

Let’s say I wanted to convert 30 miles per hour into feet per second:

30

1 ℎ ×

5280

1 ×

1 ℎ

60 ×

1

60 = 30 × 5280 ÷ 60 ÷ 60 = 44

And, of course, it works for metric units:

Convert 75 kg into cg:

75

1 ×

100

0.001 = 75 × 100 ÷ 0.001 = 7500000

It is certainly important to remember the important metric base numbers:

0.001 kX = 1 X = 10 dX = 100 cX = 1000 mX

with X representing any unit of measurement, like meters, grams, or liters.





Units for volume can be measured in cubic meter units (cm3, m3) or liter units (mL, L). These important

metric volume unit conversions should be known:

1 L = 1 dm3 or 1 mL = 1 cm3

Perform the following unit conversions. Show all work:

50 dollars = ? quarters

4500 inches2 = ? feet 2

100 miles per hour = ? inches per second (1 mi = 5280 ft)





750000 cL = ? kL 200 m = ? dm

860 mm = ? dm 600 cg = ? mg

308 m3 = ? L

64.67 dm3 = ? cL





Density

In chemistry, density is the relationship between a substance’s mass and its volume. Density is

characteristic of a given substance, and does not depend on the size of the sample. Mathematically, density

can be expressed as follows:

=

Because density is equal to mass divided by volume, the units for density are the units of mass divided by

the units for volume. Density is often expressed in grams per cubic centimeter, or grams per liter. The

following table contains the densities of some common substances. Note that different units are used to

quantify the densities of solids and liquids than are used to quantify gases.

Densities of Some Common Substances

Solids and Liquids Gases

Substance Density at 20 °C (g/cm3) Substance Density at 20 °C (g/L)

Gold 19.3 Chlorine 2.95

Mercury 13.6 Carbon dioxide 1.83

Lead 11.4 Oxygen 1.33

Aluminum 2.70 Air 1.20

Corn syrup 1.360 Nitrogen 1.17

Water (4 °C) 1.000 Neon 0.84

Corn oil 0.922 Ammonia 0.718

Ice (0 °C) 0.917 Methane 0.665

Ethanol 0.789 Helium 0.166

Gasoline 0.66 Hydrogen 0.084

Values for density can be used to identify an unknown substance.

A sample of metal has a mass of 65 grams and occupies 5.7 cubic centimeters of space. Using the table

above, identify the unknown metal.

=

=

65

5.7 = 11.4

ℎ .





Refer to the table on the previous page as you solve the following problems. Show all work.

A sample of aluminum occupies 125 cm3. What is the mass of the sample in grams?

A liquid sample occupies 48 mL, and has a mass of 37.87 g. Which of the substances in the table is it?

Neon has a density of 0.84 g/L. How many mL will be occupied by 65 g of neon?





Types of measurements

Everyone makes and uses measurements. You decide how to dress in the morning based on the measured

outside temperature. You measure out the ingredients for your favorite cookie recipe. If you were building

a cabinet, you would carefully measure each piece of wood.

Measurements are also fundamental to the experimental sciences. The reference standards used by

scientists are those of the metric system. The understanding of scientific concepts is often based on

measurements. It is important to be able to make measurements and to decide whether a measurement is

good or bad.

In chemistry, it is important to record data for every measurement you make. This data can be a recording

of numbers and units, or words. Quantitative data gives results in a definite form, using numbers and units.

Qualitative data gives a nonnumeric description.

Identify the following as quantitative or qualitative measurements.

A flame is hot.

A candle has a mass of 90 g.

Wax is soft.

A candle’s height decreases by 4.2 cm/hr.

A quantitative measurement, regardless of what it is a measurement of, is useless unless it consists of both

a number and a corresponding unit. When making a measurement, we choose a unit that makes sense

based on our perceptions and experience. When someone asks about your age, you tend to come up with

an answer in years, as opposed to seconds, days, minutes, or centuries. Similarly, if you were measuring

the height of a person, you would probably tend to choose feet or meters, rather than inches, centimeters,

or miles.

The units we will be making measurements with in this class are standard units used by scientists

throughout the world. These units are part of the International System of Units (or SI), a revised version of

the metric system. Some of the accepted SI units we will be using in this class are as follows:





Length:

The standard unit of length in the metric system is the meter. Other units of length and their

equivalents in meters are as follows:

1 millimeter = 0.001 meter1 centimeter = 0.01 meter

1 decimeter = 0.1 meter

1 kilometer = 1000 meters

For reference, 1 meter is a little longer than 1 yard or 3 feet. It is about half the height of a very tall

adult. A centimeter is nearly the diameter of a dime, a little less than half an inch. A millimeter is about

the thickness of a dime. These prefixes can be used for any metric unit.

Volume:

The standard unit of volume in the metric system is the liter. One liter is equal to 1000 cubic

centimeters in volume. 1 liter is a little more than 1 quart. One teaspoon equals about 5 milliliters.

Mass:

The standard unit of mass in the metric system is the gram. 1 gram is about the mass of a paper clip.

One kilogram is about the mass of a liter of water.

Time:

The following conversions are useful when working with time:

1 minute = 60 seconds

1 hour = 60 minutes = 3600 seconds

1 day = 24 hours

1 week = 7 days

1 year = 365 days (for the Earth to travel once around the sun)





Temperature:

Despite what you might think, energy,

temperature, and heat are not the same things.

Energy is the capacity for doing work .

Temperature is directly proportional to theaverage kinetic energy of a substance. That is,

every sample of matter is composed of tiny

particles called atoms, and they are in constant

motion. At lower temperatures, the atoms

move very slowly. At higher temperatures, they

move very rapidly.

Energy typically moves from systems at high temperatures to systems at low temperatures. The energy

that is being transferred from one body to another is known as heat , and is measured in joules (J).

You might be familiar with the Fahrenheit scale for measuring temperature. The Fahrenheit scale is not

commonly used outside the United States, especially in the scientific world. The freezing point of water

is 32°F. The boiling point of water is 212°F.

Temperature is expressed in

degrees Celsius in the metric

system. The boiling point of

water (at sea level) is 100°

Celsius, or 100°C. The freezing

point of water (at sea level) is 0°

Celsius. These numbers make

the Celsius scale convenient for

everyday use. However, the

Kelvin scale, or absolute

temperature scale, is directly

related to kinetic energy. The

zero point on the Kelvin scale

corresponds with zero

movement of particles, and is

known as absolute zero. This

occurs at 0 K, which correspondswith −273.15°C.

A change of one Kelvin degree is the same as a change of one Celsius degree. Kelvin temperatures are

always exactly 273.15 degrees higher than Kelvin temperatures.





Accuracy and precision

The words accuracy and precision are often used interchangeably, but they mean two very different things

in the world of scientific measurement. Accuracy is how close a measurement comes to the true or

accepted value of whatever is measured. Precision is concerned with the closeness of multiple

measurements to each other – the reproducibility of a measurement.

Darts stuck in a dart board can be used to illustrate the difference between these two terms. The following

four scenarios represent all possible combinations of low and high accuracy and precision.

To sum up, accuracy means getting the right answer , while precision means getting the same thing over and

over again. We will revisit these terms shortly when we cover data reporting and analysis.

Significant figures

The digits in a measurement up to and including the first uncertain digit are the significant figures of the

measurement. There are two significant figures, for example, in 62 cm3 and five in 100.00 g. The zeroes

are significant here as they signify that the uncertainty range is ± 0.01 g.

The rules for assigning significant figures are found below. They must be memorized and practiced.

ALWAYS SIGNIFICANT:

1.

Every non-zero number is a recorded measurement is significant.

13 23.45 46413 7 198.2





2.

Zeroes that occur between two non-zero numbers are significant.

202 3.02 100001 7007 30.06

3.

Zeroes all the way to the right of a decimal point are always significant.

2.0 30.00 1.0000 70.780 .300

NEVER SIGNIFICANT:

4.

Zeroes that occur to the left of non-zero numbers are NOT significant.

0.005 004 .0022 09 0.001

5.

Zeroes at the right of a number without a decimal point are NOT significant.

500 40 20000 920 6310

SOMETHING ELSE TO REMEMBER:

6.

Definitions and absolute numbers have an infinite number of significant figures.

1.000000000000000000 hour = 60.00000000000000000000 minutes

12.0000000000000000000000 eggs = 1.00000000000000000 dozen

The rules for manipulating significant figures must also be memorized.

ADDITION AND SUBTRACTION:

The answer to an addition or subtraction calculation should be rounded to the same number of decimal

places (not digits) as the measurement with the least number of decimal places.

12.52 m + 349.0 m + 8.24 m = 369.76 m → 369.8 m

2 decimal

places

1 decimal

place

2 decimal

placesAnswer should have 1 decimal place





43.253 L + 9.0245 L + 3.07 L = 55.3475 L → 55.35 L

3 decimal

places

4 decimal

places

2 decimal

placesAnswer should have 2 decimal places

MULTIPLICATION AND DIVISION:

In calculations involving multiplication and division, you need to round the answer to the same number of

significant figures as the measurement with the least amount of significant figures.

7.55 m × 0.34 m = 2.567 m2 → 2.6 m2

3 significant

figures

2 significant

figures

Answer should have 2 significant figures.

Remember to multiply units as well.

2.4526 g ÷ 8.4 cm3 = 0.291976 g/cm3 → 0.29 g/cm3

5 significant

figures

2 significant

figures

Answer should have 2 significant figures.

Remember to divide units as well.

0.7 m × 2.10 m = 1.47 m2 → 1 m2

1 significant

figure

3 significant

figures

Answer should have 1 significant figure.

Remember to multiply units as well.

Recording data and factoring in uncertainty

Measurement is an important part of chemistry. In the laboratory you will use different measuring

apparatus and there will be times when you have to select the instrument that is most appropriate for your

task from a range of possibilities. Suppose, for example, you wanted 25 cm3 of water. You could choose

from measuring cylinders, pipettes, burettes, volumetric flasks of different sizes, or even an analytical

balance if you know the density. All of these could be used to measure a volume of 25 cm3, but withdifferent levels of uncertainty.





Uncertainty in analog instruments

An uncertainty range applies to any

experimental value. Some pieces of apparatus

state the degree of uncertainty. In other cases,

you will have to make a judgment. Suppose youare asked to measure the volume of water in the

measuring cylinder shown on the right. The

bottom of the meniscus of a liquid usually lies

between two graduations and so the final figure

of the reading has to be estimated. The smallest

division in the measuring cylinder is 4 cm3 so

we should report the volume as 62 ± 2 cm3. The

same considerations apply to other equipment

such as burets and alcohol thermometers that

have analog scales. The uncertainty of ananalog scale is ± half the smallest division.

The volume reading should be taken from the

bottom of the meniscus. You could report the

volume as 62 cm3 but this is not an exact value.

Uncertainty in digital instruments

Consider a top pan balance that has a digital scale. The mass of the sample of water shown on the digital

display is 100.00 g but the last digit is uncertain. The degree of uncertainty is ± 0.01 g – the smallest scale

division. The uncertainty of a digital scale is ± the smallest scale division.

Summary

To summarize, for an analog instrument, write down all numbers that you are certain of. Then, write one

more number you estimate. Add the uncertainty, which is ± half of the smallest posted division on the

measurement apparatus.

The length of this pencil could correctly be reported as 9.49 ± 0.05 cm.

For a digital instrument, write down all numbers that are reported on the display of the instrument. Then,

add the uncertainty, which is ± the smallest scale measurement reported.









Use the rules for notation of uncertainty to answer the following:

Answer before rounding Final answer

Example: (22.2 .2) + (108.66 .05) 130.86 .25 130.9 .3

1. (44.8 .7) + (98.66 .05) = ______________ _______________

2. (88.64 .02) + (53.8 0.4) = ______________ _______________

3. (245.871 .001) + (78.88 .05) = ______________ _______________

4. (952.90 .09) – (458.34 .05) = ______________ _______________

5. (3.8 .2) + (54.67 .05) – (72.126 .005) = ______________ _______________

6. (528.11 .05) – (247.65 .02) = ______________ _______________

7. (1468.23 .05) + (3.426 .001) = ______________ _______________

8. (3.011 .001) – (1.4772 .0001) = ______________ _______________





Example (22.2 .2) x (108.66 .05) base 2412.252 max 2435.104 diff. 22.852

Answer before rounding Final Answer

2412.252 22.852 2410 20

9. (44.8 .7) x (98.66 .05) = ______________ _______________

10. (88.64 .02) (53.8 .7) = ______________ _______________

11. (245.871 .001) x (78.88 .05) = ______________ _______________

12. (952.90 .07) (458.340 .005) = ______________ _______________

13. (3.8 .2) x (54.67 .05) (.67 .04) = ______________ _______________

14. (5.26 .07) x (11.68 .05) = ______________ _______________

15. (10224.1 .5) (552.14 .05) = ______________ _______________







Examples of systematic errors:

Measuring the volume of water from the top of the meniscus rather than the bottom will lead to

volumes which are too high.

Overshooting the volume of a liquid delivered in a titration will lead to volumes which are too high.

Heat losses in an exothermic reaction will lead to smaller temperature changes.

Systematic errors can be reduced by careful experimental design. When evaluating investigations, it is

crucial that the experimenter distinguishes between systematic and random errors. You will also need to

describe the error in the conclusion. Which direction the error has gone and how it has affected your

results. Evaluation and Suggestions looks at the procedure and explains the flaws. Where did the error

come from and how can it be fixed.

Accuracy and precision and error

The smaller the systematic error, the greater will be the accuracy. The smaller the random uncertainties,

the greater will be the precision. The masses of magnesium in the earlier example are measured to the

same precision but the first set of values is more accurate.

Precise measurements have small random errors and are reproducible in repeated trials. Accurate

measurements have small systematic errors and give a result close to the accepted value.

The set of readings on the left are for high accuracy and low precision.The readings on the right are for low accuracy and high precision.

The internal assessment should have math to prove the accuracy and precision within the lab. Percent

error is used when a known value is given. A student performs a lab to find the density of aluminum. Since

there is a known value for this percent error can be used to explain the accuracy in the experiment.

Whenever multiple trials are preformed percent deviation can be used to explain how close the values are

to each other.





Accuracy is a measure of how close an experiment value is to a value which is accepted as correct. The

measure of the accuracy of an experiment value is reported as Absolute Error or Percent Error.

Absolute Error is just the difference between the measure and accepted values.

= −

Percent error or Relative Error is calculated as follows:

% = −

100%

Example

A student estimated the volume of water to be 200mL. After measuring the water with more accurate

glassware, the graduated cylinder, the water level was found to be 185mL. What was the student’s percent

error?

% = −

100%

% =200 − 185

185 100% = 8.11%

Notice that the error is a positive number if the experimental value is too high, and a negative number if the

experimental value is too low.

Frequently in science, an accepted or true value is not known. The accuracy of a measurement cannot be

reported if an accepted value is unavailable. Since scientists don‘t know how close they are toe the true

value, they repeat their experiments several times and report on how close together their values lie. It is

hoped that an experiment that can give reproducible results will also give accurate results. Certainly, it

data cannot be reproducible, it cannot be reliable.

Precision is the measure of how reproducible experimental measures are. Precision is reported as

Deviation or Difference of values.

The absolute Deviation or Absolute Difference of each measurement is the difference of each measurement

form the mean or average:

= | − |





The average deviation is the average of all the Absolute Deviations

=( )

The percent Deviation is the average Deviation reported as a percentage.

=

100%

Example

Antacid tablets were analyzed to find the amount of sodium carbonate present. The experimental values

were found to be 1.69g, 1.74g, and 1.68g. What is the percent deviation?

Mean

1.69 + 1.74 + 1.68

3= 1.70

Absolute Deviations

1: |1.69 − 1.70| = 0.01

2: |1.74 − 1.70| = 0.04

3: |1.68 − 1.70| = 0.02

Average Deviation

0.01 + 0.04 + 0.02

3= 0.02

Relative Deviation

0.02

1.70 100% = 1.2%

This tells scientists that on the average, the experiment will give values that are within 1.2% of the average

(and hopefully true) value.

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Brick 002 - Prior Knowledge - Scientific Measurement and Units