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Chapter 1: Measurements

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Chapter 1: Measurements

Study Goals

Units of measurements – the metric and SI systems

Measured numbers and exact numbers

Significant figures

Use of prefixes to change base units to larger or smaller units

Use of conversion factors

Density and specific gravity

Temperature scales – Celsius (metric) and Kelvin (SI) temperature scales

The metric system is used by scientists and health professionals throughout the world. In 1960, the System International (SI) units were adopted from the metric system by scientists to provide uniformity for units of measurements used in the sciences.

Units of Measurements – The Metric and System International (SI) Units

Unit of Length: the Meter

1 meter = 39.4 inches (in)

1 meter = 10 decimeters (dm)

= 100 centimeters (cm)

= 1000 millimeters (mm)

1 kilometer (km) = 0.621 mile

1 km = 10 hectameters

= 100 dekameters

= 1000 meters

1 US yard = 3 feet or 36 inches (hence, a meter is slightly longer than a yard.)

Metric and SI Prefixes

meters

The M-83 spiral galaxy is about 9.3 x 1019 km or 5.8 x 1019 miles away from Earth.

meter

pico p one-trillionth0.000 000 000 00110–12

femto f one-quadrillionth0.000 000 000 000 00110–15

The distance between an Oxygen atom and a Hydrogen atom in a water molecule is 95.8 pm, which is 9.58 x 10–11 m or 0.0958 nm.

Scale model of the Earth and the Moon, with a beam of light travelling between them at the speed of light (about 186,000 miles per second). The accepted Moon-Earth distance is 384,000 km or about 238,000 miles.

1 gallon = 4 quarts

1 liter (L) = 1.06 quart

1 quart = 946 mL

1 L = 10 deciliters (dL)

= 100 centilliters (cL)

= 1000 milliliters (mL)

Volume is the amount of space occupied by a substance. The metric system uses the liter (L) as the standard volume unit.

- the milliliter (mL) is commonly used for measuring smaller volumes of fluids in hospitals and laboratories.

Volumetric flasks are bottles made of glass, in a pear-like shape with long thin necks and flat bottoms. Volume marking is cut in glass around the neck. A volumetric flask is calibrated to contain (TC or In) the indicated volume of water at 20 °C. They are used for preparing the exactly known volume of sample solution and standard solutions of reagents. On each flask with volume designation a temperature on which the flask has been calibrated is designated.

Some Typical Clinical Lab Test Values

A cube measuring 10 cm on each side has a volume of 1000 cm3, or 1 L. A cube measuring 1 cm on each side has a volume of 1 cm3 (cc) or 1 ml.

A plastic intravenous fluid container contains 1000 ml.

- The mass of an object is a measure of the quantity of material it contains.
- in the metric system, the mass unit is the gram (g).
- in the SI system, the mass unit is the kilogram (kg).
- Compare to US weight units:
- 1 kg = 2.20 lbs
- 1 lb = 454 grams

Mass is not the same thing as weight. Weight has

meaning only when an object having a specific mass is placed in an acceleration field, such as the gravitational field of the earth. At the earth's surface, a kilogram mass weighs about 2.2 pounds, for example. But on Mars, the same kilogram mass would weigh only about 0.8 pounds, and on Jupiter it would weigh roughly 5.5 pounds.

The temperature of an object tells us how hot or cold the object is.

- the Celsius (C) scale is based on the temperatures of melting ice (0C) and boiling point (100C) of water as references and is divided into 100 units.

- the Fahrenheit (F) scale is also based on the temperatures of melting ice (32F) and boiling point (212 F) of water as references but is divided into 180 units.

1 C = 1.8 F

Notice that 1 degree kelvin = 1 degree Celsius

Scientific notation is used to write very large or very small numbers.

Ex: The width of a human hair of 0.000 008 m is written as 8 x 10-6 m

A large number such as 4 500 000 s is written as 4.5 x 106 s

Write scientific notation for:

(a) 93,000,000

(b) 0.00005144

(c) − 33,452.8

Solution:

(a) (2.3 x 105) (8.6 x 10−12)

= (2.3 x 8.6) x 105+(−12)

= 19.78 x 10−7

= 1.978 x 10−6

(b) 7.2 x 107 = 7.2 x 107−3

−8.23 x 103 −8.23

= −0.8748 x 104

= −8.748 x 103

Review of Algebraic Rules

- Write 6.67 x 10–11 in scientific notation with the calculator:
- First, enter the number 6.67.
- Then, press 2nd function key followed by the EE key.
- Now, enter –11.
- The display should look like this:
- 6.67 E –11

Depending on the calculator model, the display may be slightly different.

Measured numbers are numbers that in everyday's life one obtains by using measurement tools.

Examples:

a) Step on a weight scale, and a reading of 152 lbs is recorded.

b) Use a carpenter ruler to measure the length of a piece of lumber (say, 36.5 in).

- measured numbers have some built-in errors due to some degree of inaccuracy of the measuring tool as well as inaccuracy introduced by the person who performs the measuring.

Exact numbers are numbers that we obtainby counting or from a given definition.

Other examples of exact numbers:

- 1 dozen = a number equals to 12

- there are 60 seconds in 1 minute

- 1 foot = 12 inches

- 1 lb = 16 ounces

- someone asks how many siblings I have I say I have 4 siblings (= exact number).

One foot length

A dozen eggs

State whether the following are measured or exact numbers:

1. A patient weighs 155 lbs

2. The basket holds 8 apples

3. In the metric system, 1 km equals 1000 m

4. The distance from Denver, Colorado to Houston, Texas is 1720 km

5. There are 32 students in the class

6. There are 12 inches in 1 foot

7. The gravitational acceleration at sea level is 9.8 m/sec2

State whether the following numbers are measured or exact numbers and give the number of significant figures:

a) 42.2 g

b) 12 eggs

c) 0.0005 cm

d) 450 000 km

How many significant figures in the following numbers:

a) 20.60 ml

b) 1036.48 g

c) 4.00 m

d) 20.8C

e) 60 800 000 g

f) 5.0 x 10–3 L

Data from scientific measurements are often used in calculations. The number of significant figures in the measured numbers limits the numbers of significant figures that can be given in the calculated answer.

1) If the first digit to be dropped is 4 or less, it and all following digits are simply dropped from the number:

3 sig fig 2 sig fig

Example 1: 8.4234 rounded off to 8.428.4

drop 3 and all drop 2 and all

following numbers following numbers

Example 2: 3826.8 rounded off to 38303800

(note: the zeros are just place holders and are not significant)

Example 3: 2.243 x 104 rounded off to 2.24 x 104 2.2 x 104

2) If the first digit to be dropped is 5 or larger, the last retained digit of the number is increased by 1:

3 sig fig2 sig fig

Example 1: 14.780 rounded off to 14.815

drop 8 and all following drop 7 and all following

numbers; increase last numbers; increase last

retained digit 7 to 8 retained digit 4 to 5

Example 2: 0.002657 rounded off to 0.002660.0027

Example 3: 1.2856 rounded off to 1.291.3

Example 4: 24,589 rounded off to 24,60025,000

(note: the zeros are just place holders and are not significant)

Round off to four significant figures:

a) 35.7853 m =

b) 0.00262706 L =

c) 38,268 g =

d) 1.2836 kg =

In multiplification and division, the final answer has the same number of sig fig as the number with the least sig fig:

Example 1: 24.65 x 0.67 = 16.5155 17(final answer with 2 sig fig)

(4 sig fig) (2 sig fig) (calculator display)

Example 2:2.85 x 67.4 = 43.756264… 43.8(final answer with 3 sig fig)

4.39

here, all 3 numbers each has 3 sig fig, so the final value is expressed in

3 sig fig after the calculation is performed with the calculator.

- sometimes, the calculator gives a whole number, as in:

8.00 = 4 ( one must add zeros to give the

2.00 answer with correct sig fig: 4.00 )

Perform the following calculations and round off to correct significant figures:

a) 56.8 x 0.37 = c) 2.075 x 0.585 =

8.42 x 0.0045

b) 71.4 = d) 25.0 =

11 5.0

In addition and subtraction, the final answer has the same number of decimal places as the number with the fewest decimal places:

Example 1: 2.045 (3 decimal places)

+ 34.1 (1 decimal place)

36.145 rounded off to36.1(1 decimal place)

- when using a calculator, sometimes it does not give the correct sig fig number; so you must add zeros to give the answer in correct sig fig:

Example 2: 14.5 (1 decimal place)

– 2.5 (1 decimal place)

12 this is calculator's display add zero to give answer in correct

sig fig = 12.0

Perform the following calculations and give answers in correct significant figures:

a) 27.8 + 0.235 =

b) 153.247 – 14.82 =

c) 5.08 + 25.1 =

d) 85.66 + 14 + 0.025 =

e) 0.2654 – 0.2585 =

One last word on siginificant figures and calculations : When working problems, you should do the calculation with all the digits allowed by your calculator and round off only in the final answer at the end of the calculation. Rounding off in the middle of the calculation can lead to errors in the final answer.

A conversion factor is used to change one type of unit to another type of unit using an equality relationship between them.

Note: The conversion factor is an exact number relationship and is not taken

into account in terms of significant figures in the final answer.

- What is the mass inkg of a person who weighs 145 lbs ?
- The daily dose of ampicillin for the treatment of an ear infection is 115 mg/kg body weight. What is the daily dose in mg for a 34-lb child ?
- A patient needs 0.024 g of a sulfa drug. There are 8-mg tablets in stock. How many tablets should be given ?

NASA's Mars Climate Orbiter (125 million dollars) was to be inserted into orbit around Mars on 23 September 1999 after a 91/2 month journey.

Unfortunately, a failure to recognize and correct an error in a transfer of information between the orbiter spacecraft team in Colorado and the mission navigation team in California led to the loss of the spacecraft. The peer review preliminary findings indicate that one team used English units (e.g. inches, feet and pounds) while the other used metric units for a key spacecraft operation. This information was critical to the maneuvers required to place the spacecraft in the proper Mars orbit.

Sometimes a percentage is given in a problem. The term percent (%) means parts per 100 parts. In writing a percentage as a conversion factor, we choose the unit and express the numerical relationship of the parts of this unit to 100 parts of the whole.

- for example, an athlete has 18% body fat by mass. The percent quantity can be written as 18 mass units of body fat in every 100 mass units of body mass. Different mass units such as grams (g), kilograms (kg), or pounds (lb) can be used, but both units in the factor must be the same.

Example 1: John has 21% (by mass) body fat and weighs 72 kg. How much of his body weight is due to fat tissue ?

Example 2: Vinegar is about 5% (by volume) acetic acid. How many ml of acetic acid are in

a 450-ml sample of vinegar ?

A Simple Method To Test Body Fat Percentage:

Measure the Thickness of the Skin Fold at the Waist.

Density is a measure of massper unit of volume. The larger the mass of a substance relative to its volume, the denser (and heavier) it is. The density of a subtance can be calculated using the following formula:

Mass of Substance Volume of Substance

Density =

Density is an important quantitative property of matter ― it helps identify a substance since every substance has a specific density. The density of a substance indicates how its particles are packed together within the substance. The tighter the particles are packed, the higher the density of the substance.

Density of Gases

Variation with Molecular Weight

With the number of molecules constant, density varies with the molecular weight. The higher the MW, the higher the density.

Density of Gases

Variation with Temperature

A neutron star with streaming gamma rays.

The density of solid materials on Earth ranges from less than 1 g/cm3 to 22.5 g/cm3 (osmium metal). In the interior of certain stars, the density of matter is truly staggering. Black neutron stars ─ stars composed of atomic cores compressed by gravity to a superdense state ─ have densities of about 1015 g/cm3.

Density of gases depends on temperature. The higher the temperature, the lower the density; hence, warm gases rise.

The mass of a cylinder made of zinc is determined and its volume measured as shown above. The density of the zinc cylinder is therefore its mass divided by its volume.

- in the metric system, the densities of solids and liquids are usually expressed as g/cm3 or g/ml and the densities of gases in g/L. Other density units include kg/m3, g/L, and g/dm3.

Osmium ― the densest metal known. From osme, or odor; discovered 1804. A metal with a pungent smell, it is used to produce alloys of extreme hardness. Pen tips are 60% osmium. A brick-sized chunk of osmium weighs about 56 lbs.

- Mercury has a density (dHg) = 13.6 g/ml. What is the volume (ml) of mercury that contains 20.4 g mercury ?
- What is the weight of a volume of 30.6 ml mercury ?
- What is the mass of a glucose solution that fills a 0.500-L intravenous bottle if the density of the glucose solution is 1.15 g/ml ?

The diagrams below show a cube placed in a body of water. In some instances, the cube floats, while in others it sinks. Match diagrams A, B, C, or D with one of the following descriptions and explain your choices:

a) The cube has a greater density than water.

b) The cube has a density that is 0.60−0.80 g/ml.

c) The cube has a density that is 1/2 the density of water.

d) The cube has the same density of water.

Body mass is made up lean muscle, bone, body fluids, and fat tissue (adipose tissue).

- one way to determine the amount of body fat tissue is to measure the whole-body density. This can be done by underwater (hydrostatic) weighing, a standard method for estimating body fat composition.

After the on-land mass of the body is determined, the underwater body mass is obtained by submerging the person in water. Because water helps support the body by giving it buoyancy, the underwater body mass is much less.

Principle of buoyancy

A higher percentage of body fat will make a person more buoyant, causing the underwater mass to be even lower. This occurs because fat has a lower density than the rest of the body.

- the difference between the on-land mass and underwater mass is due to the buoyant force, which is equal to the mass of the water displaced by the submerged body.

- the mass of displaced water is used to determine the body volume. Then the mass and volume of the person are used to calculate his body density:

Body mass . Body volume

Body density =.

- the body density is then compared with a chart that correlates the percentage of fat tissue with body density. For example, a person with a body density of 1.05 g/ml has a 21% body fat, according to such a chart. This procedure is often used by athletes to determine exercise and diet programs.

Problem: When a 50.0-kg person is immersed in a water tank for body-fat testing, she has an underwater mass of 2.0 kg. This difference in body mass is equal to the mass of water displaced by the body. The density of water is 1.00 g/ml.

a) What is the volume (L) of water displaced ?

b) What is the volume (L) of the person ?

c) What is the body density (g/ml) of the person ?

The specific gravity is defined as the ratio of the density of a substance to the density of pure water. Thus, water is used as the reference material. The specific gravity can help quantify the buoyancy between a material relative to that of water, or determine the density of one "unknown" material using the "known" density of water.

Mathematically, specific gravity is expressed as:

Mass of Substance Volume of Substance

Specific Gravity =

- because the density of the substance and that of water have the same unit, specific gravity is a dimensionless number. If the ratio is less than 1, the substance is lighter than water. If it is more than 1, the subtances is denser than water.

- at 4C, the density of water is taken to be 1.00 g/ml.

Problem 1: the density of a vegetable oil (dvegetable oil) = 0.85 gr/ml. What is its specific gravity ?

Problem 2: John takes 2.0 teaspoons (tsp) of a cough syrup (sp gr 1.20) for a persistent cough. If there are 5.0 ml in 1 tsp, what is the mass (in grams) of the cough syrup that John took ?

The specific gravity of a subtance can be measured using a hydrometer.

1) Lead (Pb) has a density = 11.3 g/ml. A piece of lead is dropped into a graduated

cylinder, raising the water level from 215 ml to 285 ml. What is the mass (gr) of the lead ?

2) How many cubic centimeters (cm3) of olive oil have the same mass as 1.00 L of gasoline ? (given: dgasoline = 0.66 g/ml ; dolive oil = 0.92 g/ml)

3) A piece of ice weighing 35.0 g has a volume of 38.2 ml. What is the sp gr of ice ?

4) A graduated cylinder contains 18.0 ml water. What is the new water level after 35.6 g of silver metal (density = 10.5 g/ml) is submerged in the water ?

5) Mercury has a specific gravity of 13.6. How many milliliters of mercury have a mass of 0.35 kg ?