Experimental Error

1
Chapter 3

• Experimental Error

2
Experimental Error

• Experimental Error
• The uncertainty obtained in a measurement
• Results from systematic and/or random errors
• Relates to the degree of confidence in a
  measurement
• Propagation of uncertainties must be calculated
  and taken into account

3
Types of Errors

• Systematic Errors
• Sometimes referred to as determinate errors
• Arise from flaws in equipment or experimental
  design
• Reproducible with precision
• Can be discovered and corrected

4
Types of Errors

• Random Errors
• Sometimes referred to as indeterminate errors
• Arises from things that cannot be controlled
• Variations in how an individual or individuals
  read the measurements
• Instrumentation noise
• Always present and cannot always be corrected for

5
Detection of Systematic Errors

• Analyze samples of known composition
• Standard Reference material
• Develop a calibration curve
• Analyze blank samples
• Verify that the instrument will give a zero
  result
• Obtain results for a sample using multiple
  instruments
• Verifies the accuracy of the instrument
• Round Robin

6
Accuracy or Precision

• Precision
• Reproducibility of results
• Several measurements afford the same results
• Accuracy
• How close a result is to the true value
• True values contain errors since they too were
  measured
• Accuracy gets an A for correct answers

7
Calculating Errors

• Terminology
• Significant Figures minimum number of digits
  required to express a value in scientific
  notation without loss of accuracy
• Absolute Uncertainty margin of uncertainty
  associated with a measurement
• Relative Uncertainty compares the size of the
  absolute uncertainty with the size of its
  associated measurement
• Propagation of Uncertainty The calculation to
  determine the uncertainty that results from
  multiple measurements

8
Significant Figures

• How to determine which digits are Significant
• Write the number as a power of 10
• Zeros are significant and must be included when
  they occur
• In the middle of a number
• At the end of a number on the right hand side of
  the decimal point
• This implies that you know the value of a
  measurement accurately to a specific decimal point

9
Significant Figures

• Determine the number of significant digits in the
  following numbers
• 142.7 142.70
• 0.000006302 9.25 x 104
• 9.250 x 104 9.2500 x 104
• 0.3050 0.003050
• 1.003 x 104

10
Significant Figures

• The last significant digit in a measured quantity
  is the first digit of uncertainty

11
Significant Figures

• Determine the significant figures from the
  diagram below

12
Performing , -, x, or ? with Significant Figures

• The number of significant figures expressed in
  the final answer, are equal to the number of
  significant figures in the least certain number.
• The power of 10 has no effect on the number of
  significant figures expressed
• In adding or subtracting, all powers of 10 should
  have the same exponent
• Significant Figures have the exact same of
  decimal places as the of significant figures in
  the value with the least number of significant
  figures

13
Performing , -, x, or ? with Significant Figures

• The number of significant figures expressed in
  the final answer, are equal to the number of
  significant figures in the least certain number.
• All insignificant figures (those to the right of
  the least significant figure) are used to round
  the digit to the nearest significant figure
• All insignificant figures should be maintained
  until your final answer to avoid round-off error
• Commonly observed as a subscript

14
Express the answer of each of the following with
the correct of Significant Figures

• Addition and Subtraction
• 1.362 x 104 3.111 x 104
• 5.345 6.728
• 7.26 x 1014 6.69 x 1014
• 1.632 x 105 4.107 x 103 0.984 x 106
• 3.021 8.99
• 12.7 1.83

15
Express the answer of each of the following with
the correct of Significant Figures

• Multiplication and Division
• 3.26 x 10-5 x 1.78
• 4.3179 x 1012 x 3.6 x 10-19
• 34.60 ? 2.46287
• 0.0302 ? (2.1143 x 10-3)
• 6.345 x 2.2

16
Significant Figures in Logarithms and
Antilogarithms

• Logarithm of n
• n 10a or log n a
• 2 parts to a logarithm
• Characteristic integer part
• Mantissa decimal part
• Logarithm the number of significant digits
  found in n the number of significant digits in
  the mantissa
• Antilogarithm the number of significant digits
  in the mantissa the number of significant
  digits expressed in the answer

17
Express the answer of each of the following with
the correct of Significant Figures

• Logarithms and Antilogarithms
• log 339 log 1237
• log (3.39 x 10-5) log 3.2
• antilog (-3.42) antilog 4.37
• Log 0.001237 104.37
• 10-2.600 log (2.2 x 10-18)
• antilog (-2.224) 10-4.555

18
Use of Significant Figures to calculate formula
mass

• Calculate KrF2 where Kr 83.80 ? 0.01 and F
  18.9984032 ? 0.0000005
• Calculate C6H13B where C 12.0107 ? 0.0008, H
  1.00794 ? 0.00007 and B 10.811 ? 0.007

19
Relative Compared to Absolute Uncertainty

• Absolute uncertainty illustrates the uncertainty
  in a measurement
• Relative uncertainty illustrates the magnitude of
  uncertainty with regard to the measurement
• large the measurement has a large absolute
  uncertainty and therefore errors, if possible,
  need to be corrected in your instrument or
  procedure

20
Relative Compared to Absolute Uncertainty

• Relative uncertainty compares the absolute
  uncertainty with the size of the associated
  measurement
• Relative uncertainty absolute uncertainty /
  measurement
• Relative uncertainty
• relative uncertainty relative
  uncertainty x 100

21
Propagation of Uncertainty

• Since measurements commonly will contain random
  errors that lead to a degree of uncertainty,
  arithmetic operations that are performed using
  multiple measurements must take into account this
  propagation of errors when reporting uncertainty
  values

22
Propagation of Uncertainty Calculations

• Addition and Subtraction Arithmetic Functions
• Absolute uncertainty
• ey ?(ex12 ex22 )

23
What is the absolute, relative, and percent
uncertainty for the following problems?

• Addition and Subtraction
• 1.76 (?0.03) 1.89 (?0.02) 0.59 (?0.02)
• 3.4 (?0.2) 2.6 (?0.1)
• The volume delivered by a buret is the difference
  between the final reading and the initial
  reading. If the uncertainty in each reading is
  ?0.02 mL, the initial reading is 0.05 mL and the
  final reading is 17.88 mL what is the absolute,
  relative and percent uncertainty?

24
What is the absolute, relative, and percent
uncertainty for the following problems?

• Addition and Subtraction
• Express the molecular mass (? uncertainty) of
  benzene, C6H6, with the correct number of
  significant figures.
• Find the uncertainty in the molecular mass of
  B10H14 and writher the molecular mass with the
  correct number of significant figures

25
Propagation of Uncertainty Calculations

• Multiplication and Division Arithmetic Functions
• Absolute Uncertainty
• Convert all uncertainties to percent relative
  uncertainties, calculate using the following
  equation, and then convert back to absolute
  uncertainty
• ey ?(ex12 ex22 )

26
What is the absolute, relative, and percent
uncertainty for the following problems?

• Multiplication and Division
• 1.76(?0.03) x 1.89(?0.02) / 0.59(?0.02)
• 3.4(?0.2) ? 2.6(?0.1)
• 3.4(?0.2) x 10-8 ? 2.6(?0.1) x 103
• (a) A solution prepared by dissolving 0.2222
  (?0.0002) g of KIO3 FM214.0010 (?0.0009) in
  50.00 (?0.05) mL. Find the molarity and its
  uncertainty with an appropriate number of
  significant figures.
• (b) Would the answer be affected significantly
  if the reagent were only 99.9 pure?

27
What is the absolute, relative, and percent
uncertainty for the following problems?

• Multiplication and Division
• The value of the Boltzmanns constant (k) listed
  on the inside front cover of the book is
  calculated from the quotient R/N, where R is the
  gas constant (8.314472 J/(mol . K) and N is
  Avogadros number (6.02214199 x 1023/mol). If
  the uncertainty in R is ?0.000070 J/(mol . K) and
  the uncertainty in N is ?0.0000036 x 1023/mol,
  find the uncertainty in k.

28
What is the absolute, relative, and percent
uncertainty for the following problems?

• Multiplication and Division
• You prepared a 0.250 M NH3 solution by diluting
  8.45 (?0.04) mL of 28.0 (?0.5) wt NH3 density
  0.899 (?0.003) g/mL up to 500.0 (?0.2) mL.
  Find the uncertainty in 0.250 M. The molecular
  mass of NH3, 17.0306 g/mol, has negligible
  uncertainty relative to other uncertainties in
  this problem.

29
Propagation of Uncertainty Calculations

• Mixed , -, x, ? Operations
• 1. addition and subtraction functions
• 2. multiplication and division functions

30
What is the absolute, relative, and percent
uncertainty for the following problems?

• Mixed functions
• 3.4 (?0.2) 2.6 (?0.1) x 3.4 (?0.2)
• 1.76 (?0.03) 0.59 (?0.02) ? 1.89 (?0.02)

31
Propagation of Uncertainty Calculations

• Exponents and Logarithms
• Uncertainty for powers and roots
• For the equation y xa, the ey a(ex)
• Uncertainty for logarithms
• For the equation y log x, the ey (1/ln 10) x
  (ex/x)
• Do not work with relative uncertainties with
  logs and antilogs

32
Propagation of Uncertainty Calculations

• Exponents and Logarithms
• Uncertainty in Natural Logarithms
• For the equation y ln x, the ey ex/x
• Uncertainty in Antilogarithms
• For the equation y antilog x or y 10x, ey/y
  (ln 10)ex
• Uncertainty in ex
• For the equation y ex, ey/y ex

33
What is the absolute, relative, and percent
uncertainty for the following problems?

• Exponents and Logarithms
• ? 3.4 (?0.2) ? 1/2 (2)
• 3.4 (?0.2)2 ln 3.4 (?0.2)
• 103.4(?0.2) log 3.4 (?0.2)
• e3.4(?0.2)
• Consider the function pH -logH, where H
  is the molarity of H. For pH 5.21 ? 0.03,
  find H and its uncertainty.

34
Ch 3 - Homerwork

• 1, 2, 6, 7, 9, 11, 12, 15, 16, 18, 19, 20, 22, 23