CHEMISTRY 59-320CHEMISTRY 59-320ANALYTICAL CHEMISTRYANALYTICAL CHEMISTRY

Fall - 2010Fall - 2010

Lecture 4

Chapter 3 Experimental error

3.1 Significant Figures

The minimum number of digits needed to write a given value in scientific notation without loss of accuracy

A Review of Significant Figures

How many significant figures in the following examples?• 0.216 90.7 800.0 0.0670 500• 88.5470578%• 88.55%• 0.4911

The needle in the figure appears to be at an absorbance value of 0.234. We say that this number has three significant figures because the numbers 2 and 3 are completely certain and the number 4 is an estimate. The value might be read 0.233 or 0.235 by other people.

The percent transmittance is near 58.3. A reasonable estimate of uncertainty might be 58.3 ± 0.2. There are three significant figures in the number 58.3.

3.2 Significant figures in arithmetic

• Addition and subtraction The number of significant figures in the

answer may exceed or be less than that in the original data. It is limited by the least-certain one.

• Rounding: When the number is exactly halfway, round it to the nearest EVEN digit.

• Multiplication and division: is limited to the number of digits contained in the number with the fewest significant figures:

• Logarithms and antilogarithms

A logarithm is composed of a characteristic and a mantissa. The characteristic is the integer part and the mantissa is the decimal part. The number of digits in the mantissa should equal the number of significant figures.

• Problem 3-5.  Write each answer with the correct number of digits.

• (a) 1.021 + 2.69 = 3.711• (b) 12.3 − 1.63 = 10.67• (c) 4.34 × 9.2 = 39.928• (d) 0.060 2 ÷ (2.113 ×

104) = 2.84903 × 10−6

• (e) log(4.218 × 1012) = ?• (f) antilog(−3.22) = ? • (g) 102.384 = ?

• (a) 3.71  •  (b) 10.7  •  (c) 4.0 × 101 •  (d) 2.85 × 10−6 •  (e) 12.6251  •  (f) 6.0 × 10−4   • (g) 242

3-3 Types of errors

• Every measurement has some uncertainty, which is called experimental error

• Random error, also called indeterminate error, arises from the effects of uncontrolled (and maybe uncontrollable) variables in the measurement.

• Random error has an equal chance of being positive or negative.

• It is always present and cannot be corrected. It might be reduced by a better experiment.

• Systematic error, also called determinate error, arises from a flaw in equipment or the design of an experiment. It is always positive in some region and always negative in others.

• A key feature of systematic error is that it is reproducible.

• In principle, systematic error can be discovered and corrected, although this may not be easy.

Accuracy and Precision:Is There a Difference?

• Accuracy: degree of agreement between measured value and the true value.

• Absolute true value is seldom known

• Realistic Definition: degree of agreement between measured value and accepted true value.

Precision

• Precision: degree of agreement between replicate measurements of same quantity.

• Repeatability of a result

• Standard Deviation

• Coefficient of Variation

• Range of Data

• Confidence Interval about Mean Value

Illustration of Accuracy and precision.

You can’t have accuracy without good precision.

But a precise result can have a determinate or systematic error.

You can’t have accuracy without good precision.

But a precise result can have a determinate or systematic error.

Absolute and relative uncertainty:

• Absolute uncertainty expresses the margin of uncertainty associated with a measurement. If the estimated uncertainty in reading a calibrated buret is ±0.02 mL, we say that ±0.02 mL is the absolute uncertainty associated with the reading.

3-4 Propagation of Uncertainty from Random

Error • Addition and subtraction:

• Multiplication and Division: first convert all uncertainties into percent relative uncertainties, then calculate the error of the product or quotient as follows:

The rule for significant figures: The first digit of the absolute uncertainty is the last significant digit in the answer. For example, in the quotient

20.000003100 1.61045 10

0.002364 0.00005

100 0.20.025

2 221.61045 10 0.2 0.2 0.002

0.002 x 0.00946 = 0.00019

100

3-5 Propagation of uncertainty: Systematic error

• It is calculated as the sum of the uncertainty of each term

• For example: the calculation of oxygen molecular mass.

3-C.  We have a 37.0 (±0.5) wt% HCl solution with a density of 1.18 (±0.01) g/mL. To deliver 0.050 0 mol of HCl requires 4.18 mL of solution. If the uncertainty that can be tolerated in 0.050 0 mol is ±2%, how big can the absolute uncertainty in 4.18 mL be? (Caution: In this problem, you have to work backward). You would normally compute the uncertainty in mol HCl from the uncertainty in volume:

But, in this case, we know the uncertainty in mol HCl (2%) and we need to find what uncertainty in mL solution leads to that 2% uncertainty. The arithmetic has the form a = b × c × d, for which %e2

a = %e2b+%e2

c+%e2d.

If we know %ea, %ec, and %ed, we can find %eb by subtraction: %e2b = %e2a – %e2c – %e2d )

Error analysis:

0.050 0 (±2%) mol =