experimental errors & statistics

EXPERIMENTAL ERRORS & STATISTICS

NURUL AUNI BINTI ZAINAL ABIDINFACULTY OF APPLIED SCIENCE

UITM NEGERI SEMBILAN

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Scientific NotationThe number of atoms in 12 g of carbon:

602,200,000,000,000,000,000,000

6.022 x 1023

The mass of a single carbon atom in grams:

0.00000000000000000000001991.99 x 10-23

N x 10n

N is a number between 1 and 10

n is a positive or negative integer


568.762

n > 0568.762 = 5.68762 x 102

move decimal left

0.00000772

n < 00.00000772 = 7.72 x 10-6

move decimal right

Addition or Subtraction

1. Write each quantity with the same exponent n2. Combine N1 and N2 3. The exponent, n, remains the same

4.31 x 104 + 3.9 x 103 = 4.31 x 104 + 0.39 x 104 = 4.70 x 104


Multiplication1. Multiply N1 and N2

2. Add exponents n1 and n2

(4.0 x 10-5) x (7.0 x 103) = ?= (4.0 x 7.0) x (10-5+3) = 28 x 10-2 = 2.8 x 10-1

Division1. Divide N1 and N2

2. Subtract exponents n1 and n2

8.5 x 104 ÷ 5.0 x 109 = ?= (8.5 ÷ 5.0) x 104 - 9 = 1.7 x 10-5

(a x 10m) x (b x 10n) = (a x b) x 10m+n

(a x 10m) ÷ (b x 10n) = (a ÷ b) x 10m-n


Significant Figures - The meaningful digits in a measured or calculated quantity.

RULES: Any digit that is not zero is significant

1.234 kg 4 significant figures Zeros between nonzero digits are significant606 m 3 significant figures Zeros to the left of the first nonzero digit are not

significant0.08 L 1 significant figure If a number is greater than 1, then all zeros to the

right of the decimal point are significant2.0 mg 2 significant figures


If a number is less than 1, then only the zeros that are at the end and in the middle of the number are significant

0.00420 g 3 significant figures

Numbers that do not contain decimal points, zeros after the last nonzero digit may or may not be significant.

400 cm 1or 2 or 3 significant figures4 x 102 1 significant figures4.0 x 102 2 significant figures


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

24 mL 2 significant figures

3001 g 4 significant figures

0.0320 m3 3 significant figures

6.4 x 104 molecules 2 significant figures

560 kg 2 significant figures


Significant FiguresAddition or SubtractionThe answer cannot have more digits to the right of the decimal point than any of the original numbers.

89.3321.1+

90.432 round off to 90.4one significant figure after decimal point

3.70-2.91330.7867

two significant figures after decimal point

round off to 0.79


Significant FiguresMultiplication or DivisionThe number of significant figures in the result is set by the original number that has the smallest number of significant figures

4.51 x 3.6666 = 16.536366 = 16.5

3 sig figs round to3 sig figs

6.8 ÷ 112.04 = 0.0606926

2 sig figs round to2 sig figs

= 0.061


QUESTION


LOGARITHMS AND ANTILOGARITHMS

Example

Characteristics

Mantissa

log 957 = 2.981

2

0.981

log 9.57 x 10-4 = -3.019

3

0.019

In converting a number to its logarithm, the number of digits in mantissa of the log of the number (957) should be equal to the number of SF in the number (957).

10 0.072 = 1.18For antilogarithm,


Types of Errors in Chemical Analysis

1. Absolute Error

Definition: The difference between the true value and the measured value

E = xi – xt

Where xi = measured value xt = true or accepted value

Example: If 2.62 g sample of material is analyzed to be 2.52 g, so the absolute error is − 0.10g.


2. Relative Error

Definition: The absolute or mean error expressed as a percentage of the true value.

Er = xi – xt x 100% xt

The above analysis has a relative error of

− 0.10 g x 100% = -3.8% 2.62 g

* We are usually dealing with relative errors of less than 1%. A 1% error is equivalent to 1 part in 100.


2.1 Relative Accuracy

Definition: The measured value or mean expressed as a percentage of the true value.

Er = xi x 100% xt

The above analysis has a relative accuracy of

2.52 g x 100% = 96.2 % 2.62 g


3. Systematic Error or determinate error

Definition: A constant error that originates from a fixed cause, such as flaw in the design of an equipment or experiment.

It caused the mean of a set data to differ from the accepted value. This error tends to cause the results to either high every time or low every time compared to the true value. There are 3 types of systematic error:

Instrumental Error

Method Error

Personal

Error

Oct 2008


3.1 Instrumental Errors

All measuring devices contribute to systematic errors. Glassware such as pipets, burets, and volumetric flasks may hold volume slightly different from those indicated by their graduations.Occur due to significant difference in temperature from the calibration temperatere. Sources of uncertainties:

Decreased power supply voltage

Increases resistance in circuits due to temperature change


3.2 Method Errors

Non-ideal analytical methods are often sources of systematic errors. These errors are difficult to detect. The most serious of the 3 types of systematic errors.

Slow or incomplete

reaction

Instability of reacting species

Occurrence of side

reaction

Interference


3.3 Personal Errors

Involve measurements that require personal judgment. For example :

i) estimation of a pointer between tow scale divisions.

ii) color of solution.iii) level of liquids with respect to a graduation in

a burette.iv) prejudice.


3.4 Effect of Systematic Errors

•Does not change with size of the quantity measured.•Become more obvious as the size of the quantity decreases.•Approach to minimize the effect is use as large a sample as possible.

Constant Error

•Increase and decrease in proportion to the size of the sample for analysis.•Source of error : Interference due to contaminants in the sample.

Proportional Error


3.5 Detection and Control of Systematic Errors

Standard reference materials (SRM)

• There are certified samples containing a known concentration or quantities of particular analytes.

• Can be purchased from a number of governmental or industrial; sources such as U. S. National Institute of Standards and Technology (NIST).

Independent analysis

• If SRM are not available, an independent and largely different analysis can be used in parallel with the method evaluated.

• A statistical test must be used to determine whether the difference is due to random errors in the 2 methods.

Analysis of blank sample

• Blank contains the reagents and solvents used in analysis but no analyte.

• Reveals errors due to interfering contaminants from the reagents and vessels used in analysis.


4 Random Error or Indeterminate error

Cause data to be scattered more or less symmetrically around a mean value.

It reflects the precision of the measurement.

This error is caused by the many uncontrollable variables in physical or chemical measurements.


5 Gross Error

Differ from indeterminate and determinate errors.

They usually occur only occasionally, may cause a result to be either high or low.

For example:i) part of precipitate is lost before weighing,

analytical results will be low.ii) touching a weighing bottle with your fingers

after empty mass will cause a high mass reading for a solid weighed.

Lead to outliers, results in replicate measurements that differs significantly from the rest of the results.

experimental errors & statistics

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