Chapter 1 Chemistry and You ‘SI Units of Measure and Uncertainties

Chapter 1 Chemistry and You

‘SI Units of Measure and Uncertainties'

SI Units and Uncertainties

SI Unit (Le Système International d’Unités)

Fundamental units meter (m) kilogram (kg) second (s) ampere (A) Kelvin (K) mole (mol) candela (cd)

SI Units and Uncertainties

Derived Units Any unit made of 2 or more

fundamental units m s-1

m s-2

Newton (N) = kg m s-2

Joule (J) = kg m2 s-2

Watt (W) = kg m2 s-3

Coulomb (C) = A s

Estimation with SI Units

Fundamental Units Mass: 1 kg – 2.2lbs / 1 L of H2O /

An avg. person is 50 kg Length: 1 m - Distance between one’s

hands with outstretched arms Time: 1 s - Duration of resting heartbeat

Derived Units Force: 1 N- weight of an apple Energy: 1 J- Work lifting an apple off of

the ground

Scientific Notation and Prefixes

SI prefixes Table

1 Gm = 1,000,000,000 m = 1,000,000 km1 GM = 1 x 109 m = 1 x 106 km

0.0000000001 s = 1 ?s = ? ms

Uncertainties & Errors

A. Random Errors1. Readability of an instrument2. A less than perfect observer3. Effects of a change in the

surroundings

Can be reduced by repeated readings

B. Systematic Errors1. A wrongly calibrated instrument2. An observer is less than perfect

for every measurement in the same way

Cannot be reduced by repeated readings

Uncertainties & Errors (cont.)

•An experiment is accurate if……it has a small systematic error

it has a small random error

x

x

x

x

Systematic error

Random errors

Perfect

•An experiment is precise if……

Uncertainties & Errors (cont.)

Accuracy and Precision:

Precise but not accurate

Accurate but not precise

Precise and accurate!

Precision– uniformityAccuracy- conformity

to a standard

Determining the Range of Uncertainty

1) Analogue scales (rulers,thermometers meters with needles)

±half of the smallest division

2) Digital scales

±the smallest division on the readout

If the digital scale reads 5.052g, then the uncertainty would be ± 0.001g

10

40

30

20

50

Since the smallest division on the cylinder is 10 ml, the reading would be 32 ± 5 ml

Absolute Uncertainty- has units of the measurement

Range of Uncertainty (cont.)

3. Significant Figures

•The measurement is 14.742 g, the uncertainty of the measurement is 14.742 ± .001 g•The measurement is 50ml, the uncertainty of the measurement is 50 ± 1 ml

If you are given a value without an uncertainty, assume its uncertainty is ±1 of the last significant figure

Examples:

Range of Uncertainty (cont.)

4. From repeated measurements (an average)

Find the deviations between the average value and the largest and smallest values.

Example: A student times a cart going down a ramp 5 times, and gets these numbers: 2.03 s, 1.89 s, 1.92 s, 2.09 s, 1.96 s Average: 1.98 s

The average is the best value and the largest deviation is taken as the uncertainty range:

Largest: 2.09 - 1.98 = 0.11 sSmallest: 1.98 - 1.89 = 0.09 s

1.98 ± 0.11 s

Mathematical Representation of Uncertainty

Find the density of a block of wood if its mass is 15 g ± 1 g and its volume is 5.0 ± 0.3 cm3

= g5.0 cm3

= 3.0 g cm-3

For calculations, compare the calculated value without uncertainties (the best value) with the max and min values with uncertainties in the calculation.

Example 1:

Best value

mv

Density =

Mathematical Representation of Uncertainty

Find the density of a block of wood if its mass is 15 g ± 1 g and its volume is 5.0 ± 0.3 cm3

= g4.7 cm3

= 3.40 g cm-3

Example 1 (cont.):

Maximum value:

mv

Density =

Minimum value:

mv

Density = = g5.3 cm3

= 2.64 g cm-3

Mathematical Representation of Uncertainty (cont.)

•The uncertainty in the previous problem could have been written as a percentage

In this case, the density is 3.0 g cm-3 ± 13%

yy

= 3

X 100% = 13%

•The uncertainty range of our calculated value is the largest difference from the best value..

In this case, the density is 3.0 ± 0.4 g cm-3

Mathematical Representation of Uncertainty (cont.)

Example #2: What is the uncertainty of cos if = 60o ±5o?

•Best value of cos = cos 60o = 0.50•Max value of cos = cos 55o = 0.57•Min value of cos = cos 65o = 0.42

The largest deviation is taken as the uncertainty range:

In this case, it is 0.50 ± .08 OR 0.50 ± 16%

Deviates 0.07

Deviates 0.08

Mathematical Representation of Uncertainty: Shortcuts!

When 2 or more quantities are added or subtracted, the overall uncertainty is equal to the sum of the individual uncertainties.

Addition and Subtraction:

y = a + b Uncertainty of 2nd quantity

Uncertainty of 1st quantity

Total uncertainty

Mathematical Representation of Uncertainty: Shortcuts! (cont.)

•Determine the thickness of a pipe wall if the external radius is 4.0 ± 0.1 cm and the internal radius is 3.6 ± 0.1 cm

Example for Addition and Subtraction:

Internal radius = 3.6 ± 0.1 cm

External radius = 4.0 ± 0.1 cm

Thickness of pipe: 4.0 cm – 3.6 cm = 0.4 cm

Uncertainty = 0.1 cm + 0.1 cm = 0.2 cm

Thickness with uncertainty: 0.4 ± 0.2 cm OR 0.4 cm ± 50%


The overall uncertainty is approximately equal to the sum of the percentage (or fractional) uncertainties of each quantity.

Multiplication and Division:

y = a + b + cy a b c Denominators

represent best values

Total percentage/ fractional uncertainty

Fractional Uncertainties of each quantity


Using the density example from before (where the mass was 15 g ± 1 g and its volume is 5.0 ± 0.3 cm3)

Example for Multiplication and Division:

y = a + by a b

= 1 + 0.3

15 5= 0.07 + 0.06 = .13 ( this means 13%)

13% of 3 g cm-3 is 0.4 g cm-3

3.0 ± 0.4 g cm-3 or 3.0 g cm-3 ± 13%

The result of this calculation with uncertainty is:


Just multiply the exponent by the percentage (or fractional) uncertainty of the number.

For exponential calculations (x2, x3):

Cube- each side is 6.0 ± 0.1 cm Example:

Percent uncertainty

= 1.7%0.16

x 100 %=

Volume = (6 cm)3 = 216 cm3

Uncertainty in value = 3 (1.7%) = ± 5.1% (or 11 cm3)

Therefore the volume is 216 ± 11 cm3

Problems:

1. If a cube is measured to be 4.0+_ 0.1 cm in length along each side.

Calculate the uncertainty in volume.

Answer: 64+_5 Cm

Problem ( IB 2010)

The length of each side of a sugar cube is measured as 10 mm with an uncertainty of +_2mm. Which of the following is the absolute uncertainty in the volume of the sugar cube?

a.+_6 mm c. +_400 mmb. +_8 mm d. +_600 mm

Problem:

3. The lengths and width of a rectangular plates are 50+_0.5 mm and 25+_0.5 mm. Calculate the best estimate of the percentage uncertainty in the calculated area.

a. +_0.02% c. +_3%b. +_1 % d. +_5%

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Chapter 1 Chemistry and You ‘SI Units of Measure and Uncertainties