Math Session: - Measurement - Dimensional Analysis

SC155: Introduction to ChemistryFreddie Arocho-Perez

Numbers in Science

• Integer Numbers: 2 6

• Signed Numbers:-2 +4

• Irrational/Decimal Numbers:2.3 0.35 -2.68 +9.87

English vs. Metric System

Physical Quantity Metric Unit English Unit

Mass Gram (g) Pound (lb)

Volume Liter (L) Gallon (gal)

Length Meter (m) Inch (in)

Time Second (s) Minute (min)

Temperature Celsius (°C)Kelvin (K)

Fahrenheit (°F)

Metric System• Length

– Measurement of distance or dimension.– The base unit: meter.– It is a little over 1 yard long, more precisely 39.4 inches long. Here

are some other conversions:1 meter (m) = 39.4 inches = 1.094 yards (about one big step)1 meter (m) = 100 centimeters (cm)1 kilometer (km) = 1000 meters = 0.62 miles

• Mass– Amount of matter or material in an object.– The base unit: gram.– Here are some other conversions:

1 gram (g) = 0.0353 ounce 1 pound (lb) = 453.6 g1 ounce (oz) = 28.35 grams1 kilogram (kg) = 1000 grams

Metric System• Volume

– Amount of space occupied by an object.

– The base unit:• liter (L)• milliliter (mL)

– 1 L = 1,000 mL– A milliliter is a cube 1 cm

long on each side (1 cm3).– 1 mL = 1 cm3 = 1 cc

Temperature

Temperature

• In scientific measurements, the Celsius (C) and Kelvin (K) scales are most often used.

• The Celsius scale is based on the properties of water.– 0 C is the freezing point of water– 100 C is the boiling point of water

Temperature• Kelvin is one of the standard units of

temperature:

K = C + 273.15• Celsius is the other standard unit.• Fahrenheit is not used in scientific

measurements.• Other Formulas:

F = (1.8 x C) + 32

C = (F - 32) x 0.555

Temperature• If a weather forecaster predicts that the

temperature for the day will reach 31 C, what is the predicted temperature: (a) in K ?

(b) in F ?

• Solution:– (a) Using Kelvin Equation, we have

K = C + 273.15

= 31 + 273.15

= 304.15 K ~ 304 K

Temperature• Temperature: 31 C

• Solution:– (b) Using Fahrenheit Equation, we have

F = (1.8 x C) + 32

= (1.8 x 31) + 32

= 55.8 + 32

= 87.8 F

~ 88 F

Temperature

• 85.0 F is approximately the same as?

• Solution: Use the Celsius Equation

C = (F - 32) x 0.555

= (85.0 - 32) x 0.555

= 53 x 0.555

= 29.4 C

Density

• Physical property of a substance

• Relation between mass and volume

v

m

volume

massd

Density

• Calculate the density of mercury if 100 g occupies a volume of 7.36 mL.

• Solution:

d = m / v

d = 100 g / 7.36 mL

d = 13.6 g/mL

volume

massDensity

Dimensional Analysis

• Dimensional Analysis– Also called Factor-Label Method or

the Unit Factor Method

• This a problem-solving method that uses the fact that any number or expression can be multiplied by one without changing its value.

Dimensional Analysis

• Unit factors may be made from any two terms that describe the same or equivalent “amounts” of what we are interested in.

• For example, we know that:

1 inch = 2.54 centimeters

1 dozen = 12 items

Dimensional Analysis

• We can make two unit factors from this information:

1 dozen = 12 items

OR

12 items = 1 dozen

• Arrangement:

unitgiven

unitdesiredunitgivenunitdesired

Dimensional Analysis

• How many items are in 2 dozens?

• Conversion Factor: 1 dozen = 12 items

• Solution:

itemsdozen

itemsdozensitems 24

1

122?

Dimensional Analysis

• How many dozens are in 6 items?

• Conversion Factor: 1 dozen = 12 items

• Solution:

dozenitems

dozenitemsdozens 5.0

12

16?

Dimensional Analysis

• How many centimeters are in 6.0 inches?

• Conversion Factor: 1 in = 2.54 cm

• Solution:

cminch

cminchescm 2.15

1

54.20.6?

Dimensional Analysis

• How many inches are 24.0 centimeters?

• Conversion Factor: 1 in = 2.54 cm

• Solution:

inchescm

inchcminches 4.9

54.2

10.24?

Dimensional Analysis

• Convert 5.0 L to milliliters (mL).

• Conversion Factor: 1 L = 1,000 mL

• Solution:

mLL

mLLmL 000,5

1

000,10.5?

Dimensional Analysis

• Convert 50.0 mL to liters (L).

• Conversion Factor: 1 L = 1,000 mL

• Solution:

LmL

LmLL 05.0

000,1

10.50?

Dimensional Analysis

• If a lady has a mass of 115 lb, what is her mass in grams?

• Answer: 52,164 g

Solution: Because we want to change from lb to g, we look for a relationship between these units of mass. We have that 1 lb = 453.6 g. In order to cancel pounds and leave grams, we write the conversion factor with grams in the numerator and pounds in the denominator:

Dimensional Analysis

• You can also string many unit factors together. • How many minutes are in 2.0 years?

= 1,051,200 minutes

hourday

hours

yr

daysyr

1

min60

1

24

1

3650.2min?

Dimensional Analysis

• Units are a critical part of describing every measurement.

• Before you can work with units mathematically, you frequently must convert from one unit to another.

• Dimensional analysis does not do your math for you, but it makes sure you get your multiplications and divisions straight.

• After that, all you have to do is find the conversion factors and plug into a calculator.

Significant Figures

• The term significant figures refers to digits that were measured.

• When rounding calculated numbers, we pay attention to significant figures so we do not overstate the accuracy of our answers.

Significant Figures

1. All nonzero digits are significant.

2. Zeroes between two significant figures are themselves significant.

3. Zeroes at the beginning of a number are never significant.

4. Zeroes at the end of a number are significant if a decimal point is written in the number.

Significant Figures

• Examples: How many significant figures are present in the following numbers?

Number # Significant Figures Rule(s)

48,923 5 1

3.967 4 1

900.06 5 1, 2

0.0004 1 1, 3

8.1000 5 1, 4

Significant Figures

• When math operations are performed, answers are rounded to the number of digits that corresponds to the least number of significant figures in any of the numbers used in the calculation.

• Example: How many significant figures should be shown for the following calculation?

• Answer: 0.621799561 = 0.62 (2 significant figures)

1.25 0.45

2.734

Powers of Ten • Scientific Notation

• Way to deal with large and small numbers: abbreviate them.

• Examples:0.00001 = 1 x 10-5

0.005 = 5 x 10-3

3,000 = 3 x 103

100,000 = 1.0 x 105

6,000,000 = 6.0 x 106

Powers of Ten• For numbers larger than 10, the power of 10 is a positive value and

negative for numbers less than 1.• For numbers between 0 and 10, the power is a positive fraction.• In the examples that follow, notice what happens to the decimal point:

100 = 1. = 1. with the decimal point moved 0 places

101 = 10. = 1. with the decimal point moved 1 place to the right

102 = 100. = 1. with the decimal point moved 2 places to the right

106 = 1000000. = 1. with the decimal point moved 6 places to the right

And

10-1 = 0.1 = 1. with the decimal point moved 1 place to the left

10-2 = 0.01 = 1. with the decimal point moved 2 places to the left

10-6 = 0.000001 = 1. with the decimal point moved 6 places to the left