Measurements and Calculations

1
Chapter 2

• Measurements and Calculations

2
Scientific Notation Review

• One number to the left of the decimal point
• Practice
• 87839 _________
• 0.000327 _________
• 5.8 x 102 _________
• 4.789 x 10 3 _________

3
Units

• Numbers without units are meaningless!
• In science classes, we use the metric system
• Typical metric units
• Length m, cm, nm
• Mass g, mg, kg
• Volume mL, L

4
The uncertainty of your measurements is affected
by your equipment and your lab technique

• Uncertainty is a function of
• Precision
• Accuracy

5
How do you determine equipment precision?

• The more decimal places to the RIGHT of the
  decimal point, the more precise the equipment is.
• Generally, more precise equipment is more
  expensive.

6

• Watch from Target 15
• Reads hours and minutes, but not seconds
• The atomic clock at NIST costs 9 billion
• Reads the time as 13.3153874532
• The best cesium oscillators (such as NIST-F1) can
  produce frequency with an uncertainty of about
  10-15, which translates to a time error of about
  0.1 nanoseconds per day.

7
Electronic Balances A 1.0 g B 1.01 g
C 1.001 g The last number is the most
uncertain value (it may bounce around when youre
trying to read it)The balance that is most
precise is

• Balance A
• Balance B
• Balance C

8
What is the volume of liquid in this beaker?

• The smallest division is 10 mL, so we can read
  the volume to 1/10 of 10 mL or 1 mL. The volume
  we read from the beaker has a reading error of 1
  mL.
• The volume in this beaker is 47 mL. You might
  have read 46 mL your friend might read the
  volume as 48 mL. All the answers are correct
  within the reading error of 1 mL.
• So, how many significant figures does our volume
  of 47 mL have? Answer - 2! The "4" we know for
  sure plus the "7" we had to estimate (the
  uncertainty).

9
What is the volume of liquid in the graduated
cylinder?

• 36 mL
• 36.5 mL
• 36.4 mL
• 36.6 mL
• B,C or D

10
What is the volume which has been used up in this
buret? Note the increasing value of the numbers
from top of buret to bottom.

• 20.3 mL
• 21.6 mL
• 20.38 mL

11
How do you determine instrument accuracy?

• Accuracy of the equipment depends on the
  instruments calibration.
• Example calibrate a balance
• place a standard weight of known mass on the
  balance and observing the balances reading.
• An accurate balance will have a reading the same
  as the standard mass, within the equipments
  precision.
• NIST (National Institute for Science and
  Technology) provides standard weights to use for
  calibration.

12
What about the effect of the experimenterlab
ratyou?

• Precision
• how close several measurements of the same item
  are to each other telling the same story over and
  over again
• being consistent
• Accuracy
• How close the measurements are to the known value
• the Truth

13

• A B
  C
• The dartboard that is precise but not accurate
  is
• The dartboard that is accurate but not precise
  is
• The dartboard that is precise AND accurate is

14
There are two ways to calculate accuracy

• Chemist lingo
• error experimental theoretical x 100
• theoretical
• Physicist lingo
• error measured expected x 100
• expected value

15
Significant Figures

• The two kinds of numbers in the world are
• Exact numbers
• Values obtained by COUNTING
• Inexact numbers
• Values obtained by MEASURING
• Significant numbers include all digits that are
  certain plus the first uncertain number

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• All digits are significant in an exact number,
  because the number is obtained by counting
• 12 eggs in a dozen
• 5 digits on a hand
• The number of significant figures in an inexact
  (measured) number depends on the precision of the
  measuring device
• One ruler may give a reading of 220 mm (2 sig
  figs)
• A more precise ruler may give a reading of 221 mm
  (3 sig figs)
• A really precise ruler may give a reading of
  221.3 mm (4 sig figs)

17
It is common sense that the answer to a
calculation can be known no more precisely than
the least precise piece of information.
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How to determine the number of significant figures

• Zeros are the big problem
• Leading zeros are never significant.
• Imbedded zeros are always significant
• Trailing zeros are significant only if the
  decimal point is specified
• Hint Write the number in scientific notation.

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• Example Sci. Not. Sig Figs
• 0.00682 6.82 x 10-3 3
• Leading zeros are not significant.
• 1.072 1.072 x 100 4
• Imbedded zeros are always significant.
• 300 3 x 102 1
• Trailing zeros are significant only if the
  decimal point is specified.
• 300. 3.00 x 102 3
• 300.0 3.000 x 102 4

20
How many significant figures are in each of the
following?

• 1.23 ____ 250.
  ______
• 0.00123 _____ 2.50 x 102 ______
• 2.0 ____ 3.228 x 10-5
  ______
• 0.020 ______ 2.0005 x 101 _____
• 100 ______
• 100. ______
• 100.00 ______
• 250 ______

21
Rules for working with sig figs

• Addition and subtractionThe number of decimal
  places in the answer should be equal to the
  number of decimal places in the with the fewest
  places.
• Add 0.12 g (2 decimal places)
• 1.6 g (1 decimal place)
• 10.976 g (3 decimal places)
• 12.696 g
• 12.7 g (1 decimal place)

22
Find the average age of the following students

• 17. years old
• 16.8
• 15.9
• 18
• 18
• 85.7 / 5 17.14 years old
• Add and then divide by exact number and then
  limit answer to the least number of decimal
  places.
• 17.14 Round before you chop off
• 17 limit

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Rules for working with sig figs

• Multiplication and divisionThe number of sig.
  figures in the answer should be the same as that
  in the quantity with the fewest significant
  figures.
• Multiply 0.01208 (4 sig figs)
• x .0236 (3 sig figs)
• 0.000285088 (calculator answer)
• 0.000285
• 2.85 x 10-4

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Practice!

• The correct numbers of significant figures for
  the values 4500, 4500., 0.0032 and 0.04050 are
• 4, 4, 4, 5
• 2, 2, 2, 2
• 2, 4, 2, 4

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• The correct answer for the sum
• 4503 34.90 550
• A. 5088
• B. 5090
• C. 5100

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• Give the answer to the correct number of
  significant figures
• 1.367 - 1.34
• A. 0.027
• B. 0.030
• C. 0.03

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• Give the answer to the correct number of
  significant figures
• (1.3 x 103)(5.724 x 104)
• A. 7.4 x 107
• B. 7.441 x 107
• C. 7.44 x 107

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• Give the answer to the correct number of
  significant figures
• (6305)/(0.010)
• A. 6.30500 x 105
• B. 6.31 x 105
• C. 6.3 x 105

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Dimensional Analysis is the way to ensure that
your units are converted correctly

• How many seconds are there in one day?
• Convert 8160 A to nanometers
• (1nm 10 A)
• NOTE conversions are EXACT numbers

30

• What is the mass of the snow, in tons, on a 150
  ft by 45 ft flat roof after a 6.0 in. snowfall?
  Assume that 11 inches of snow is equivalent to
  1.0 inch of water.
• density 1.0 g/cm3,
• 1 lb 454 g,
• 1 ton 2000 lbs
• HINT Find the area of the roof, and then the
  volume of snow on the roof

31
Temperature Scales

• Deg. F is only used in the USA now!
• The Kelvin scale allows you to report temperature
  without using negative numbers

32
Converting from F to C

• Boiling to freezing in F is 212-32 180
• Boiling to freezing in C is 100-0 100
• So 1 deg. C 1.8 deg.F OR 1 deg. F 4/9 deg.C
• 4/9(deg.F 32) deg.C
• ExampleYour temperature in France is 39.5 deg.C.
  What will you tell your parents in America?

33
Converting from deg.C to K

• K deg. C 273
• NOTE you never use the deg. symbol when
  reporting in K!
• You have attained a temperature of 0 K. What is
  this temperature in deg.C?

34
Density m/V

• Materials can be differentiated based on their
  density (gold vs. lead)
• Three calculations
• Given mass and volume, find density
• Given density and mass, find volume
• Given density and volume, find mass

35
Density Review

• What is the mass in grams of a 9.00 cm3 piece of
  lead (d 11.3 g/cm3)?
• A. 101.7 g
• B. 102.
• C. 102. g

36
Density Review

• A 250 cm3 volume of a liquid weighs 312. g. What
  volume of the liquid will weigh 4.5 g?
• A. 3.61 cm3
• B. 3.6 cm3
• C. 5.6 cm3