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Chapter 3: Scientific Measurement

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Chapter 3: Scientific Measurement

- 3.1 Measurements and Their Uncertainty
- 3.2 The International System of Units
- 3.3 Conversion Problems
- 3.4 Density

Last modified 8-06

Corresponds with Prenice Hall text Ch. 3

3.1 Measurements and Their Uncertainty

On January 4, 2004, the Mars Exploration Rover Spirit landed on Mars. Each day of its mission, Spirit recorded measurements for analysis. In the chemistry laboratory, you must strive for accuracy and precision in your measurements.

Using and Expressing Measurements

A measurement is a quantity that has both a number and a unit.

Measurements are fundamental to the experimental sciences. For that reason, it is important to be able to make measurements and to decide whether a measurement is correct.

Scientific Notation

In scientific notation, a given number is written as the product of two numbers: a coefficient and 10 raised to a power.

The number of stars in a galaxy is an example of an estimate that should be expressed in scientific notation.

“Writing a number as a power of 10.”

Why? It makes very large and very small numbers more manageable to write and use.

- Rule of thumb: Use when number is greater than 1000 or smaller than 0.001. Or, you may always use it!
The number of sig. figs are clearly shown in a measurement.

Scientific Notation

How important is a change in the power of 10?

Diameter of Earth’s orbit around the sun

≈ 100,000,000,000 m = 1.0*1011 m

Diameter of an atom

≈ 0.0000000001 = 1.0*10-10 m

Writing in scientific notation

- Move the decimal point in the original number so that it is located to the right of the first nonzero digit.
- Multiply the new number by 10 raised to the proper power that is equal to the number of places the decimal moved.
- If the decimal point moves:
- To the left, the power of 10 is positive.
- To the right, the power of 10 is negative.

Accuracy, Precision, and Error

Accuracy is a measure of how close a measurement comes to the actual, true, or accepted value of whatever is measured.

Precision is a measure of how close a series of measurements are to one another.

- How closely individual measurements compare with each other
- The “fineness” of a measurement
To evaluate the accuracy of a measurement, the measured value must be compared to the correct value. To evaluate the precision of a measurement, you must compare the values of two or more repeated measurements.

Accurate or Precise?

- What is the temperature at which water boils?
- Measurements: 95.0°C, 95.1°C, 95.3°C
- True value: 100°C

Precise!

(but not accurate)

Accurate or Precise?

- What is the temperature at which water freezes?
- Measurements: 1.0°C, 1.2°C, -5.0°C
- True value: 0.0°C

Accurate!

(it’s hard to be accurate without being precise)

Accurate or Precise?

- What is the atmospheric pressure at sea level?
- Measurements: 10.01 atm, 0.25 atm, 234.5 atm
- True value: 1.00 atm

Not Accurate & Not Precise

(don’t quit your day job)

Accurate or Precise?

- What is the mass of one Liter of water?
- Measurements: 1.000 kg, 0.999 kg, 1.002 kg
- True value: 1.000 kg

Accurate & Precise

(it’s time to go pro)

Accuracy, Precision, and Error

Accuracy, Precision, and Error

Determining Error

The accepted value is the correct value based on reliable references.

The experimental value is the value measured in the lab.

The difference between the experimental value and the accepted value is called the error.

- The percent error is the absolute value of the error divided by the accepted value, multiplied by 100%.

Accuracy, Precision, and Error

Ex. Find the percent error if your data shows 1.23 g Mg and the true answer is 1.40 g Mg.

Accuracy, Precision, and Error

Just because a measuring device works, you cannot assume it is accurate. The scale below has not been properly zeroed, so the reading obtained for the person’s weight is inaccurate.

Accuracy or Precision?

When deciding on accuracy, precision, both, or neither….it is quantitative data (numerical), not qualitative (descriptive)

- The recipe calls for 25 chocolate chips per cookie. The cookies analyzed have 34, 35, and 32 respectively.
- The percent NaCl is 99%, 99%, and 98%.
- The number of grams of KF required is 0.04 g. The amounts used were 0.038, 0.039, 0.041, and 0.040.
- To win, Henry must earn 500 points. In his three trials, he earned 400, 480, and 395 points.

41.2 mL (3 sig figs = very precise)

41.0

50 mL Graduated cylinder

100 mL Beaker

A beaker:

40. mL (2 sig figs = not as precise)

50

Significant Figures in Measurements

The significant figures in a measurement include all of the digits that are known, plus a last digit that is estimated.

Ex. Suppose you estimate a weight that is between 2.4 lb and 2.5 lb to be 2.46 lb. The first two digits (2 and 4) are known. The last digit (6) is an estimate and involves some uncertainty. All three digits convey useful information, however, and are called significant figures.

Significant Figures in Measurements

Significant Figures

Every measurement has some degree of uncertainty.

Significant figures (“sig figs”): the digits in a measurement that are reliable (or precise). The greater the number of sig figs, the more precise that measurement is.

A more precise instrument will give more sig. figs. in its measurements.

- To measure the time for a pencil to fall…compare a stopwatch and a wall clock.
- To measure the volume of a liquid…compare a graduated cylinder and a beaker.

The stopwatch & graduated cylinder are more precise instruments…so the readings they produce will have more sig figs.

Significant Figures in Calculations

In general, a calculated answer cannot be more precise than the least precise measurement from which it was calculated. The calculated value must be rounded to make it consistent with the measurements from which it was calculated.

To round a number, you must first decide how many significant figures your answer should have. The answer depends on the given measurements and on the mathematical process used to arrive at the answer.

PACIFIC

The “Atlantic-Pacific” Rule“PACIFIC”

Decimal point is PRESENT. Count digits from left side, starting with the first nonzero digit.

40603.23 ft2

= 7 sig figs

0.01586 mL

= 4 sig figs

ATLANTIC

When are digits “significant”?“ATLANTIC”

Decimal point is ABSENT. Count digits from right side, starting with the first nonzero digit.

3 sig figs =

40600 ft2

1 sig fig =

1000 mL

Examples

- 0.00932
Decimal point present → “Pacific” → count digits from left, starting with first nonzero digit

= 3 sig figs

- 4035
Decimal point absent → “Atlantic” → count digits from right, starting with first nonzero digit

= 4 sig figs

- 27510
Decimal point absent → “Atlantic” → count digits from right, starting with first nonzero digit

= 4 sig figs

Sig. Figs. In Calculations And Scientific Notation

- In this class, we will not follow the sig. fig. rules for operations

- In this class, use scientific notation for all numbers greater than 1000 and smaller than 0.001

- In this class, we will carry all digits in our calculations and then round the final answer to 3 significant digits

Exception…in very large or very small numbers, you may use 4 digits in intermediate steps (i.e. 0.0034567432 can be rounded to 0.003457)

Write the following measurements in scientific notation, then record the number of sig figs.

- 789 g
- 96,875 mL
- 0.0000133 J
- 8.915 atm
- 0.94°C

3 sig figs

7.89*102 g

5 sig figs

9.6875*104 mL

3 sig figs

1.33*10-5 J

4 sig figs

8.915 atm

2 sig figs

9.4*10-1 °C

3.1 Section Quiz

1. In which of the following expressions is the number on the left NOT equal to the number on the right?

a) 0.00456 10–8 = 4.56 10–11

b) 454 10–8 = 4.54 10–6

c) 842.6 104 = 8.426 106

d) 0.00452 106 = 4.52 109

3.1 Section Quiz

2. Which set of measurements of a 2.00-g standard is the most precise?

a) 2.00 g, 2.01 g, 1.98 g

b) 2.10 g, 2.00 g, 2.20 g

c) 2.02 g, 2.03 g, 2.04 g

d) 1.50 g, 2.00 g, 2.50 g

3.1 Section Quiz

3. A student reports the volume of a liquid as 0.0130 L. How many significant figures are in this measurement?

a) 2

b) 3

c) 4

d) 5

3.2 The International System of Units

In the signs shown here, the distances are listed as numbers with no units attached. Without the units, it is impossible to communicate the measurement to others. When you make a measurement, you must assign the correct units to the numerical value.

Measuring with SI Units

All measurements depend on units that serve as reference standards. The standards of measurement used in science are those of the metric system.

The International System of Units (abbreviated SI, after the French name, Le Système International d’Unités) is a revised version of the metric system.

Measuring with SI Units

The five SI base units commonly used by chemists are the meter, the kilogram, the kelvin, the second, and the mole.

Units and Quantities

Units of Length

In SI, the basic unit of length, or linear measure, is the meter (m). For very large or and very small lengths, it may be more convenient to use a unit of length that has a prefix.

History of SI

- The metric system or Système International d'Unités (S.I.), was first organized in Paris as part of the French Revolution and adopted by France in 1795. At that time, the meter and kilogram were standardized.
- Why was it organized? It is simple, being based on powers of 10. (like our number system)
- Every country in the world uses the metric system except the USA, Myanmar, and Liberia.
- By 2009, all products sold in Europe must use the metric system. No dual-labelling will be permitted.
- Visit U.S. Metric Association (USMA), Inc. for more info.

Units and Quantities

Common metric units of length include the centimeter, meter, and kilometer.

Units and Quantities

Units of Volume

- The SI unit of volume is the amount of space occupied by a cube that is 1 m along each edge. This volume is the cubic meter (m)3. A more convenient unit of volume for everyday use is the liter, a non-SI unit.
- A liter (L) is the volume of a cube that is 10 centimeters (10 cm) along each edge (10 cm 10 cm 10 cm = 1000 cm3 = 1 L).
- Common metric units of volume include the liter, milliliter, cubic centimeter, and microliter.

Units and Quantities

- A sugar cube has a volume of 1 cm3. 1 mL is the same as 1 cm3.

The volume of 20 drops of liquid from a medicine dropper is approximately 1 mL.

Units and Quantities

Units of Mass

- The mass of an object is measured in comparison to a standard mass of 1 kilogram (kg), which is the basic SI unit of mass.
- A gram (g) is 1/1000 of a kilogram; the mass of 1 cm3 of water at 4°C is 1 g.
- Common metric units of mass include kilogram, gram, milligram, and microgram.

Units and Quantities

Weight is a force that measures the pull on a given mass by gravity.

The astronaut shown on the surface of the moon weighs one sixth of what he weighs on Earth.

Units and Quantities

Units of Temperature

- Temperature is a measure of how hot or cold an object is.
- Thermometers are used to measure temperature.

Units and Quantities

- Scientists commonly use two equivalent units of temperature, the degree Celsius and the kelvin.
- On the Celsius scale, the freezing point of water is 0°C and the boiling point is 100°C.
- On the Kelvin scale, the freezing point of water is 273.15 kelvins (K), and the boiling point is 373.15 K.
- The zero point on the Kelvin scale, 0 K, or absolute zero, is equal to 273.15 °C.

- °F = (1.8 * °C) + 32
- °C = (°F-32)
1.8

Units and Quantities

Units of Energy

Energy is the capacity to do work or to produce heat.

- The joule (J) is the SI unit of energy.
- One calorie (cal) is the quantity of heat that raises the temperature of 1 g of pure water by 1°C.
- One Calorie (Cal) equals 1000 calories

Units and Quantities

- This house is equipped with solar panels. The solar panels convert the radiant energy from the sun into electrical energy that can be used to heat water and power appliances.

3.2 Section Quiz.

1. Which of the following is not a base SI unit?

a) meter

b) gram

c) second

d) mole

3.2 Section Quiz.

2. If you measured both the mass and weight of an object on Earth and on the moon, you would find that

a) both the mass and the weight do not change.

b) both the mass and the weight change.

c) the mass remains the same, but the weight changes.

d) the mass changes, but the weight remains the same.

3.2 Section Quiz.

3. A temperature of 30 degrees Celsius is equivalent to

a) 303 K.

b) 300 K.

c) 243 K.

d) 247 K.

3.3 Conversion Problems

Because each country’s currency compares differently with the U.S. dollar, knowing how to convert currency units correctly is very important. Conversion problems are readily solved by a problem-solving approach called dimensional analysis.

Conversion Factors

A conversion factor is a ratio of equivalent measurements.

The ratios 100 cm/1 m and 1 m/100 cm are examples of conversion factors.

Conversion Factors

When a measurement is multiplied by a conversion factor, the numerical value is generally changed, but the actual size of the quantity measured remains the same.

The scale of the micrograph is in nanometers. Using the relationship 109 nm = 1 m, you can write the following conversion factors.

Dimensional Analysis

Dimensional analysis is a way to analyze and solve problems using the units, or dimensions, of the measurements.

Problems in which a measurement with one unit is converted to an equivalent measurement with another unit are easily solved using dimensional analysis.

Otherwise known as factor label method

Dimensional Analysis

Use “railroad tracks”

Show units

Box answer or write answer in statement unless part of a larger problem

YOU MUST DO IT MY WAY AND SHOW WORK IF YOU WANT CREDIT

Ex. Convert 140,500,000 mL to kL

140,500,000 mL L kL = 140.5 kL (141 kL)

1000 mL 1000L

Practice problems

- 1) Convert 55°F to °C and K
- 2) How many seconds in a year?
- 3) Convert 1.23 km to m and then to um
- 4) Convert 42.8 kg to oz.

3.3 Section Quiz

2. The conversion factor used to convert joules to calories changes

a) the quantity of energy measured but not the numerical value of the measurement.

b) neither the numerical value of the measurement nor the quantity of energy measured.

c) the numerical value of the measurement but not the quantity of energy measured.

d) both the numerical value of the measurement and the quantity of energy measured.

3.3 Section Quiz

3. Express the density 5.6 g/cm3 in kg/m3.

a) 5.6 106kg/m3

b) 5.6 103kg/m3

c) 0.56 kg/m3

d) 0.0056 kg/m3

3.4 Density

If you think that these lily pads float because they are lightweight, you are only partially correct. The ratio of the mass of an object to its volume can be used to determine whether an object floats or sinks in water.

Determining Density

Density is the ratio of the mass of an object to its volume.

Density is an intensive property that depends only on the composition of a substance, not on the size of the sample.

Determining Density

- Each of these 10-g samples has a different volume because the densities vary.

The density of corn oil is less than the density of corn syrup. For that reason, the oil floats on top of the syrup.

Experiments show that the volume of most substances increases as the temperature increases. Meanwhile, the mass remains the same. Thus, the density must change. The density of a substance generally decreases as its temperature increases.

What is density?

- The density of water is 1.0 g/mL (At 4.0°C).
- The relative density of any substance can be determined by adding it to water.
- If the substance floats, it’s density is less than the density of water:
d < 1.0 g/mL

- If the substance sinks, it’s density is greater than the density of water:
d > 1.0 g/mL

0.80 g/mL

1.0 g/mL

1.2 g/mL

Practice problems

- What is the density of water?
- What is the density of ice?
- Calculate the density of a liquid if 40.0 mL of the liquid has a mass of 90.2 g. Is the liquid more or less dense than water?
- If the liquid were added to a beaker of water, what would happen?

1.0 g/mL at 4°C

0.917 g/mL

2.26 g/mL; more

Don’t know—could sink to the bottom or it could mix

with the water (polar vs. nonpolar solutions)

3.4 Section Quiz

1. If 50.0 mL of corn syrup have a mass of 68.7 g, the density of the corn syrup is

a) 0.737 g/mL.

b) 0.727 g/mL.

c) 1.36 g/mL.

d) 1.37 g/mL.

3.4 Section Quiz

2. What is the volume of a pure gold coin that has a mass of 38.6 g? The density of gold is 19.3 g/cm3.

a) 0.500 cm3

b) 2.00 cm3

c) 38.6 cm3

d) 745 cm3

3.4 Section Quiz

3. As the temperature increases, the density of most substances

a) increases.

b) decreases.

c) remains the same.

d) increases at first and then decreases.

What you need to know: Chapter 3

3.3

conversion factor

dimensional analysis

railroad tracks

3.4

density—definition

density—water

3.1

scientific notation

precision vs. accuracy

% error

significant figures (identifying)

rounding (3 sig. fig.)

3.2

SI base units (meter, kilogram, kelvin, second, mole)

metric prefixes—know values (kilo, centi, milli)

metric prefixes—recognize relative size (giga, mega, deci, micro, nano, pico)

temperature (K, C, F); absolute zero

Last modified 9-07

Practice Problems

- Write problems in your notebook. Head the top of the page with “Ch. 3 HW” followed by the page and numbers assigned. Write in complete sentences/answers for credit.
- pp. 96-97
- (57-59, 63-64, 66, 68, 70-71, 74-75, 78, 82, 85, 87)

The Metric Prefixes

Extra Slides….

What is a meter?

- *In the 1790’s, a meter was originally defined as “1/10,000,000th of the distance from the Equator to the North Pole measured along the meridian that runs through Dunkirk, France.”
- *It is currently defined as the “distance through which a beam of light travels in a vacuum in 1/299,792,458 of a sec.”
- (It’s also exactly 39.37 inches.)

What is a Liter?

- *The S.I. unit for volume is actually the cubic meter (m3) but that’s pretty big.
- The Liter is defined as a cube measuring 1 decimeter on each side, or 1 dm3, or 1000 cm3.
- *How many Liters are there in 1 m3?
- *So, the Liter is based on the meter, which is based on the Earth.

1 dm

1dm

1 dm

*What is a kilogram?

- The S.I. unit for mass is the kilogram, and is defined as the mass of 1 dm3 of water at 4°C.
- Why water?
- So, the kilogram is based on the Liter, which is really based on the meter, which is really based on the Earth.

1 dm

1dm

1 dm

*What is a second?

- The S.I. unit for time is the second. It was originally defined as 1/86,400th of the average solar day.
- It is presently defined with some high level chemistry/physics concepts. (1 second is the time that elapses during 9.192631770 x 109 cycles of the radiation produced by the transition between two levels of the Cs-133 atom.)
- So, the second is based on the time it takes the Earth to rotate.

What is a kelvin? *So, the kelvin is based on water and absolute zero (the coldest temperature possible.)

- *The S.I. unit for temperature is the kelvin, and is defined as 1/100 of the temperature difference between the boiling point & freezing point of water at one atmosphere of pressure.
- The kelvin (K) and the degree Celsius (°C) are exactly the same size, although 1 degree Fahrenheit (°F)is equal to about 1.8°C.
- To convert:
- K = °C + 273.15
- °F = (1.8 * °C) + 32
- °C = (°F-32)
1.8