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Measurements and Units Accuracy, Precision, and Errors: Measurements and Units

Measurements and Units Accuracy, Precision, and Errors

A measurement is a quantitative observation that consists of two parts: a number and a unit. Why are units important? 

 

45,000

  has little meaning – just a number

45,000 dollars

  has some meaning – money

45,000 dollars/year

  has more meaning – a person’s salary

 

Scientists realized long ago that standard systems of units had to be adopted if measurements were to be useful and they were to communicate their results. For this  reason, the SI System (International System of Units - Système Internationale d’Unités)  was established as the scientific system of measurement. These are some fundamental  quantities in the SI system:

 

 

Physical Quantity

Unit

Abbreviation

Length

meter

m

Mass

kilogram

kg

Time

second

s

Temperature

kelvin

K

Number of particles 

mole

mol

Electric current

ampere

A

Luminous intensity

candela

cd

 

 

All other physical quantities have units that can be derived from the SI base units.

e.g. velocity = displacement / Δ time - Derived SI units: m s-1

e.g. acceleration = Δ velocity / Δ time - Derived SI units: m s-2

 

SI is a decimal system. Quantities differing from the base unit by powers of ten are noted by use of prefixes. Changing the prefix alters the size of a unit.

e.g. 1 kilometer = 1000 m. The most common SI prefixes can be found in any first year chemistry or physics textbook. 

                                                                           © cryptosmith.com

 

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Dimensional Analysis

It is often necessary to convert a given result from one SI unit to another (e.g. km to m). We use dimensional analysis and conversion factors to get the result in the units we want. 

Consider: 1 m = 100 cm

Thus, the ratio 100cm/1 m is a conversion factor that, when multiplied by any length in meters, converts that length to centimeters.

4.2m \times \frac{{100cm}}{{1m}} = 420cm

 

Essentially, conversion factors are the ratio between two units.

It is often necessary to convert a given result from one SI unit to another (e.g. km to m). We use dimensional analysis and conversion factors to get the result in the units we want. 

 

 

To convert a quantity with one set of units to a different set of units, we set up a conversion pathway by multiplying by one or more conversion factors.


 

EXAMPLE: Creatinine is a substance found in blood. In an analysis of a blood serum sample that detected 0.58 mg of creatinine, how many micrograms are present? 

 

 

0.580mg \times \frac{{1g}}{{1000mg}} \times {\rm{}}\frac{{1 \times {{10}^6}\mu g}}{{1g}} = 580\mu g

 

There are cases where the units of both the numerator and denominator must be converted. 

 

 

EXAMPLE: A nerve impulse in the body can travel as fast as 400 feet/second. What is its speed in meters/ min? Given: 1 m = 3.3 ft.

Desired units  \frac{m}{{min}} = \frac{{400ft}}{{1s}} \times {\rm{}}\frac{{1m}}{{3.3ft}} \times \frac{{60s}}{{1min}} = 7273\frac{m}{{min}}

Cancelling out units is a way of checking out that your calculation is set up correctly. Use the same method for more complicated situations (when units are squared, cubed, etc.). 

EXAMPLE: How many square feet (ft 2) correspond to an area of 1.00 square meter (m 2 ) given that 1 m = 39.37 in and 12 in = 1 ft.? 

f{t^2} = 1.00{m^2} \times {\left( {\frac{{39.37in}}{{1m}}} \right)^2} \times {\left( {\frac{{1ft}}{{12in}}} \right)^2} = 10.8f{t^2}

 

Conversion factors can be raised to other powers. 

 

 

f{t^2} = 1.0{m^2} \times {\rm{}}\frac{{{{\left( {39.37} \right)}^2}i{n^2}}}{{1{m^2}}}{\rm{}} \times {\rm{}}\frac{{1f{t^2}}}{{{{\left( {12} \right)}^2}i{n^2}}} = 10.8f{t^2}

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Measuring Properties of Matter

Basically, the properties of matter can be classified into two groups: extensive and intensive properties.

Extensive properties depend on the quantity of the sample observed, such as mass and volume. 

Mass (m) is the quantity of matter in an object. The SI base unit is the kilogram (kg); however, the gram (g) is more commonly used in Chemistry. The mass of an object is fixed and independent of where or how it is measured.

Weight is the force of gravity on an object, directly proportional to mass: W ∝ m and W=mass  \times  Gravity  . The weight of an object may vary because gravity varies slightly from one point on Earth to the other.

 

Mass

Weight

Mass is always a constant (at any place and time)

Gravity changes and  depends on gravity at the place

Is measured in Kilograms (SI system)

Is measured in Newtons

Can never be zero

Can be Zero ( with no gravity)

Is an intrinsic property of an object and is dependent of any external factor

Depends on:

- Mass of the object

- Force and gravity

                               

 

Basically, the properties of matter can be classified into two groups: extensive and intensive properties.

Extensive properties depend on the quantity of sample observed, such as mass and volume. 

 

Volume is the amount of space that an object occupies. The more common non-SI unit to use is the liter (L). The most common unit used in the lab is the milliliter (mL). Recall that:

 

1 mL = 1 cm3 (cubic centimeter)

 

1 L = 1 dm3 (cubic decimeter) 

 

Intensive properties do not depend on the quantity of sample observed; they are characteristics of the substance itself being measured, regardless of quantity, and can be used to identify substances. Some examples of intensive properties are temperature, density, melting and boiling points. 

Temperature is the measure of the thermal energy of a system. Three common scales are used: Fahrenheit (°F), Celsius/Centigrade (°C), and Kelvin (SI temperature scale). The Kelvin (K) scale assigns a value of zero to the lowest conceivable temperature, 0 K unit ( -273.15 °C). 

 

EXAMPLE: Converting °F to °C

 If the temperature outside is 20 °F, what is the temperature in °C? 

T\left( {^\circ {\rm{C}}} \right) = \left[ {T\left( {^\circ {\rm{F}}} \right)} \right] - 32]{\rm{}} \times {\rm{}}\frac{{5}}{{9{\rm{}}}}

T\left( {^\circ {\rm{C}}} \right) = \left[ {20^\circ {\rm{F}} - {\rm{}}32} \right]{\rm{}} \times {\rm{}}\frac{{5}}{{9{\rm{}}}} =  - 6.7^\circ {\rm{C}}

 

EXAMPLE: Converting between Kelvin and Celsius Scales

 If the temperature outside is 20.10 °C, what is the temperature in K? 

T\left( {\rm{K}} \right) = T\left( {^\circ {\rm{C}}} \right) + 273.15

T\left( {^\circ {\rm{C}}} \right) = T\left( {\rm{K}} \right) - 273.15

T\left( {\rm{K}} \right) = 20.10^\circ C + 273.15 = 293.25Kunit

Note some common temperatures:

  • 0 °C = 273.15 K (often rounded off to 273 K)
  • 0 K = -273.15 °C (often rounded off to -273 °C )
  • 25 °C = 298.15 K (often rounded off to 298 K)

Intensive properties do not depend on the quantity of the sample observed; they are characteristics of the substance itself being measured, regardless of quantity, and can be used to identify substances.

Density (d) is an intensive property of a substance based on two extensive properties: mass and volume.

 

 

Density\left( d \right) = \frac{{mass\left( m \right)}}{{volume\left( V \right)}}

 

SI derived unit: kg/m3. More common units used in the lab are g/cm3 and g/mL 

Intensive properties do not depend on the quantity of the sample observed; they are characteristics of the substance itself being measured, regardless of quantity, and can be used to identify substances.

Density is a function of temperature because volume varies with temperature, whereas mass remains constant. 

d\left( {{H_2}O} \right){\rm{ }} = {\rm{ }}1.000{\rm{ }}g/mL  at 4 °C

d\left( {{H_2}O} \right){\rm{ }} = {\rm{ }}0.9982{\rm{ }}g/mL  at 20 °C

 

EXAMPLE: What is the density of 5.00 mL of a fluid if it has a mass of 5.23 g?

d = \frac{m}{v}

d = \frac{{5.23g}}{{5.00mL}} = 1.05\frac{g}{{mL}}

EXAMPLE: What would be the mass of 1.00 L of this sample?

m= V × d

m = 1.00L \times {\rm{}}1000\frac{{mL}}{L} \times {\rm{}}1.05\frac{g}{{mL}} = 1.05 \times {10^3}g

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Credits

Created by peer tutors under the direction of Learning Centre faculty at Douglas College, British Columbia.

 

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