CHAPTER 1: Measurements

Study Goals:

• Learn the units and abbreviations for the metric (SI) system.
• Distinguish between measured numbers and exact numbers.
• Determine the number of significant figures in a measurement.
• Use prefixes to change base units to larger or smaller units.
• Write conversion factors from the units in an equality.
• In problem-solving, convert the initial unit of a measurement to another unit.
• Round off a calculator answer to report an answer with the correct number of significant figures.
• Calculate temperature values in degrees Celsius and Kelvin.

1.1 Units of Measurement

• We make measurements every day.
• For instance, you make a measurement every time you:
  • Measure your height
  • Weigh produce
  • Take your temperature
• In every measurement is followed by a unit
  • 7 ft. 5 inches
  • 2.5 pounds
  • 98.3 degrees Fahrenheit
• Why start with measurements?
• Science and allied health professionals measure and report many quantities.
  • How do are these measurements used?
• Comparison of measurements
  • In order to compare measurements, we must report them in terms of a unit. e.g, lb. (weight), ft., in (height), etc
  • Measurements are often compared to a standard
  • Units are the standard of measurement.
• Scientists have adopted a common set of units to make comparison of measurements easier.
  • International System of Units or Système International (SI)
    • based on metric system
    • metric system is 10-base - each unit is divided into 10 smaller units.

• Length - figure
  • SI Unit = meter (m)
  • 1 (m) = 39.4 (in)
  • centimeter (cm) is convenient unit for measuring smaller distances
• Volume
  • SI Unit = cubic (m³)
  • 1 (L) = 1.06 (qt)
  • milliliter (mL) is convenient unity for measuring smaller amounts of liquid
• Mass
  • SI Unit - gram (kg)
  • 1 kg = 2.20 lb
    • 454g = 1 lb
  • Weight and mass are different concepts.  Mass does not vary and weight is dependent on the gravitational pull
• Temperature
  • SI Unit - K
• • Thermometer with both metric (°C) and U.S. (°F) scales
  • Practice with problems 1.2, 1.3, 1.5 in your book

1.2 Scientific Notation

• Scientific notation is used for very large and very small numbers
• e.g.
  • 0.000008 → 8 X 10^-6
  • 100,000 → 1 X 10⁵
• Powers of 10 - exponent
  • 1000 = 10 X 10 X 10 X 10 = 1 X 10³ (3 is the power of 10)
• 2 parts
  • coefficient
  • exponent (power of ten)
• Converting Scientific Notation to a Standard Number
  • Power of ten determines the number of spaces to move the decimal point.
  • Negative - decimal is moved to the left
  • Positive - decimal is moved to the right

1.3 Measured and Exact Numbers

• Measured Numbers
• Known & Estimated Digits
• Measurements always include a degree of uncertainty
  • The measurement contains certain numbers and estimated numbers
  • The last number is uncertain
  • If the measurement lands on a number, a zero is added to indicate the uncertainty
• For example;
  • In the length measurement of 2.75 cm,
  • the digits 2 and 7 are certain (known)
  • the third digit 5(coluld be 6 or 7) is estimated (uncertain)
  • all three digits (2.75) are significant including the estimated digit.
• Exact Numbers
  • Exact numbers do not have any uncertainty
  • Defined relationships
    • 60 seconds in a minute
  • Counting Numbers
    • 28 students in class
  • Exact numbers are not obtained with a measuring tool.

1.4 Significant Figures

• Reported measurements depend upon measuring tool
• Measurements are not exact
• Significant figures for a measurement include all known digits plus one estimated digit.

Precision and Accuracy

Precision: how well measured quantities agree with each other Accuracy: how well measured quantities agree with the true value The first target on the left has good accuracy and good precision - all arrows are in the bull's-eye. The second target has poor accuracy but good precision - all arrows are clustered together but far from the bull's-eye. The third target has poor accuracy and poor precision - the arrow are spread apart and not close to the target What would a target look like that had good accuracy and poor precision?

Significant Figures

• When scientists and allied health professional make measurements, they often use them in calculations
• How does the result of a calculation reflect the precision of the measured number?
• In a measurement it is important to indicate the exactness of the measurement. This exactness is reflected in the number of significant figures
• Guidelines for determining the number of significant figures in a measured quantity are:
  • The number of significant figures is the number of digits known with certainty plus one uncertain digit. (Example: 2.2405 g means we are sure the mass is 2.240 g but we are uncertain about the nearest 0.0001 g.)
  • Final calculations are only as significant as the least significant measurement.
• Rules:
  1. Nonzero numbers and zeros between nonzero numbers are always significant
  2. Zeros before the first nonzero digit are not significant. (Example: 0.0003 has one significant figure.)
  3. Zeros at the end of the number after a decimal point are significant
  4. Zeros at the end of a number before a decimal point are ambiguous (e.g., 10,300 g). Exponential notation eliminates this ambiguity
• Method:
  1. Write the number in scientific notation
  2. The number of digits remaining is the number of significant figures
  3. Examples:

  2.50 x 10² cm has 3 significant figures as written
  1.03 x 10⁴ g has 3 significant figures
  1.030 x 10⁴ g has 4 significant figures
  1.0300 x 10⁴ g has 5 significant figures.

1.5 Significant Figures in Calculations

• Multiplication and division:
  • The answer is written so it has the same number of decimal places as the measurement having the fewest decimal places
    (e.g., 6.221 cm x 5.2 cm = 32 cm²).
• Addition and subtraction:
  • The final answer is written so it has the same number of digits as the measurement with the fewest significant figures
    (e.g., 20.4 g 1.322 g = 19.1 g).
• In multiple step calculations always retain an extra significant figure until the end to prevent rounding errors.
• • Rounding Off
    1. If the first digit to be dropped is 4 or less, it and all following digits are simply dropped from the number
    2. If the first digit to be dropped is 5 or greater, the last retained digit of the number is increased by 1

Counting significant figures

• Try the significant figure tutorial on the disk that came with your book.
• The significant figures are all the reported numbers including the estimated digit.
• Zeros may or may not be significant depending on their position.
• All nonzero numbers are counted as significant figures.
• Leading zeros are decimal place holder.  Leading zeros are not significant
• Sandwiched zeros occur between nonzero numbers. Sandwiched zeros are significant.
• Trailing zeros follow non-zero numbers.  This situation is ambiguous.  Trailing zeros are usually placeholders are are not significant, but sometimes they are significant.

Measurement	Number of Significant Figures	Scientific Notation
38.15 cm	4	3.815 x 10-1 cm
5.6 ft	2	5.6 x 10-1 ft
65.6 lb	3	6.56 x 10-1 lb
100.55 m	5	1.0055 x 10-1 m
0.0005 cm	1	5 x 10-4 cm
450 000 km	??	4.5?? x 10-5 km

Significant Figures in Scientific Notation

• All digits in the coefficient are significant
• Writing a number in scientific notation makes it easier to determine the significant figures!!

1.6 SI and Metric Prefixes

SI Units

• 1960: All scientific units use Systeme International d'Unites (SI Units).
• There are seven base units.

• Smaller and larger units are obtained by decimal fractions or multiples of the base units.

Length and Mass

• SI base unit of length = meter (1 m = 1.0936 yards).
• SI base unit of mass (not weight) = kilogram (1 kg = 2.2 pounds).
  • Mass is a measure of the amount of material in an object.

Derived SI Units

• These are formed from the seven base units
• Example: Velocity is distance traveled per unit time, so units of velocity are units of distance (m) divided by units of time (s): m/s.

Volume

• S.I. Units of volume = (units of length)³ = m³
• This unit is unrealistically large, so we use more reasonable units:
  • cm³ [also known as mL (milliliter) or cc (cubic centimeters)]
  • dm³ (also known as liters, L)
• Important: the liter is not an SI unit. Derived SI Unit is m³
• Conversion: converting a cubed value? Remember to cube the conversion factor!

Density

• Is used to characterize substances
• Density is defined as mass divided by volume
• Units: g/cm³ or g/mL (for solids and liquids); g/L (often used for gases)
• Was originally based on mass (the density was defined as the mass of 1.00 g of pure water at 25°C).
• Prefixes are attached to a unit to increase or decrease its size by a factor of 10.
  • e.g 1 kg = 1000 g
  •       1 km = 1000 m
  • (these relationships are called equalities)

1.7 Problem Solving Using Conversion Factors (Dimensional Analysis)

• Dimensional analysis is a method of calculation utilizing a knowledge of units
• Given units can be multiplied and divided to give the desired units
• Conversion factors are used to manipulate units
  • desired unit = given unit x (conversion factor)
• The conversion factors are simple ratios.
  • conversion factor = (desired unit) / (given unit)
    • These are fractions whose numerator and denominator are the same quantity expressed in different units.
    • Multiplication by a conversion factor is equivalent to multiplying by a factor of one.
• Equalities - the quantity in an equality use two different unit values to describe the same measured amount
• An equality can be written to derive conversion factors.
• e. g:
  • 1 m = 1000 mm
  • 1 lb = 16 oz
  • 2.20 lb = 1 kg

Exact and Measured Numbers in Equalities

• Equalities written between units of the same system are definitions; they are exact numbers
• Equalities written between units of different systems represent measured numbers and must be treated as significant figures

Conversion Factors

• A conversion factor is a fraction where the quantities in an equality are written as a numerator and denominator
  • e.g equality - 1 hour. = 60 min

Numerator Denominator

60 min 1 hour and 1 hour 60 min

Problem Solving with Conversion Factors

• Quantity (Initial Unit) x Conversion factor = Same quantity (New Unit)

Metric Conversion Factors

• Recall that the metric system is a 10 base system
• All metric to metric conversions of the same unit are multiples of 10
• e.g

1 m = 100 cm

⇒

100 cm 1 m and 1 m 100 cm

Metric-U.S Conversion Factors

• Conversion between the U.S. system to the metric system
• e.g

1 kg = 2.20 lb

⇒

1 kg 2.20 lb and 2.20 lb 1 kg

Problem Solving

• A word problem may contain information that can be used to write conversion factors

At the store the price of a pound of peppers is $2.39		At the gas station, one gallon of gas is $3.04
1 lb $2.39	$2.39 1 lb	1 gal $3.04	$3.04 1 gal

• Solving a Problem
  • A person has a height of 2 meters, what is that height in inches
• Step 1:  Determine initial and final units
  • Initial = meters (m)
  • Final = inches (in)
• Step 2:  Determine quantity in initial units
  • 2 meters
• Step 3: Turn the equality into conversion factors

1 m = 39.4 in

⇒

1 m 39.4 in and 39.4 in 1 m

• Step 4:  Setup the Problem so that the units cancel:

2 m

X

39.4 in 1 m

=

78.8 in

• The unit left is the final unit

Using Two or More Conversion Factors

• Often two or more conversion factors are required to obtain the desired unit
• Additional unit conversion factors are placed in the problem setup to cancel preceding units

Unit 1

X

Unit 2 Unit 1

X

Unit 3 Unit 2

=

Unit 3

• Example: How many minutes are in 1.4 days?

1.4 days	X	24 hr 1 day	X	60 min 1 hr	=	2.0 x 103 min
Initial Unit		1st Conversion Factor		2nd Conversion Factor

• Always check your unit cancelation!!

Clinical Calculations Using Conversion Factors

• Conversion factors are used for calculating dosages - see problems in text p 25

Using Percents as Conversion Factors

• A percent refers to a ratio of the parts to the whole

%

=

Parts Whole

X

100

Summary of Dimensional Analysis

In dimensional analysis always ask three questions:

1. What data are we given?
2. What quantity do we need?
3. What conversion factors are available to take us from what we are given to what we need?

1.8 Density

• Density compares the mass of an object to its volume
• the mass of an object of substance is written in the numerator and its volume in the denominator:

Density = mass volume

⇒

g mL or g cm3

= g/cm3

• Note 1 mL = 1 cm³
• Density is determined by volume displacement
• the units for density are unit mass/ unit volume
• common units are g/mL

Density as a conversion Factor

• Consider a substance with a density of 3.8 g/mL

3.8 g = 1 mL

⇒

3.8 g 1 mL and  1 mL 3.8 g

Specific gravity (sp gr)

• Specific gravity is a ratio between the density of a substance and the density of water.
  • Specific gravity = (density of sample / density of water)
• Specific gravity is unitless
• e.g.  The density of mercury is 13.6 g/mL, What is the specific gravity of mercury?

Specific Gravity

=

density of mercury density of water

=

13.6 g/mL 1.0 g/mL

=

13.6

1.9 Temperature

• Temperature is the measure of the hotness or coldness of an object.
• Scientific studies use Celsius and Kelvin scales
• Celsius scale: water freezes at 0°C and boils at 100°C (sea level)
• Kelvin scale (SI Unit): Water freezes at 273.15 K and boils at 373.15 K (sea level)
  • is based on properties of gases
  • Zero is the lowest possible temperature (absolute zero)
  • 0 K = 273.15°C
• Fahrenheit (not used in science):
  • Water freezes at 32°F and boils at 212°F (sea level)

• Converting between Celsius and Fahrenheit:
  • The freezing and boiling temperatures of water are used as reference points on the temperature scale
  • On the F scale there are 180 degrees between waters freezing and boiling points
  • On the C scale there are 100.

180 Fahrenheit units = 100 Celsius units

⇒

180 Fahrenheit units 100 Celsius units = 1.8 Fahrenheit units 1 Celsius unit

• To convert a Celsius temperature to Fahrenheit the value of 32 is used to adjust the zero point (water Freezing) to 32° on the Fahrenheit scale.
• Convert °C to °F

°F

=

1.8°F 1°C

(°C)

+

32°

• Convert °F to °C

°C

=

(°F - 32°) 1.8

• A person with hypothermia has a body temperature of  34.8°C.  What is that temperature in °F?
• 94.6°F

Kelvin Temperature Scale

• On the Kelvin scale the lowest possible temperature has been adjusted to zero
• °K and °C scales have the same divisions
  • i.e., 100 degrees between the boiling and freezing points of water
• To convert °C into °K
• K = °C + 273