Download Result of a measurement = number x unit

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Chemistry 4
Chapter 3
Measurement and Chemical Calculations
1 INTRODUCTION TO MEASUREMENT
Measurement of a quantity usually can be made by comparison with a standard that serves to
represent the magnitude of a unit.
Result of a measurement = number x unit
There are three major systems of measurement units in wide use: The US Customary System, The
British Imperial System, and the International System (SI) ( sometimes called Metric System).
The SI units are used almost exclusively for scientific work.
2 EXPONENTIAL (SCIENTIFIC) NOTATION
A number, n, may be written in exponential notation as follows:
n = c x 10e
Where c is the coefficient. In scientific notation ( Standard exponential notation ) the coefficient is
equal to or greater than 1 and less than 10 : 1 < c < 10
10e is an exponential.
e is the exponent. It is an integer and may be positive or negative.
Procedure for writing number in exponential notation:
1) Rewrite the number, placing the decimal after the first nonzero digit.
2) Count the number of places the decimal in the original number moved to its new place.
3) Compare the original number with the coefficient in step 1
If the coefficient is smaller, the exponent is positive
If the coefficient is larger, the exponent is negative. Insert a minus sign in front of the
exponent
(The exponent is positive if the decimal moves left, and negative if it moves right )
For example
34567890 = 3.456789 x 107
0.0000534 = 5.34 x 10-5
To multiply or divide in exponential notation, work with coefficients and exponentials separately
and then combine. Remember that
10a x 10b = 10a+b
10a / 10b = 10a-b
For example
(3.96 x 104 ) ( 5.19 x 10-7 ) = (3.96 x 5.19 ) ( 104 x 10-7)
= 20.6 x 10-3
= 2.06 x 10-2
To add or subtract exponential numbers, adjust all exponentials to same power, then add or subtract
coefficient, the exponent of the result is the same as those of the numbers being added or subtracted.
2.24 x 10-2 + 1.12 x 10-4 = 2.24 x 10-2 + 0.0112 x 10-2
= (2.24 + 0.0112 ) x 10-2
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= 2.25 x 10
-2
3 DIMENSIONAL ANALYSIS
Many chemical problems involving quantities that are directly proportional can be solved by a method
called dimensional analysis. When two quantities are directly proportional, we can find conversion factors
to calculate the value of one property from the value of the other property.
Information required = information given x conversion factor
Example 1. Let us consider the problem: How many days are there in 3 weeks?
The steps to solve this problem are:
1) Identify the given quantity: 3 weeks
2) Identify the wanted quantity: ? days
3) Find the conversion factor. From equality
7 days ≡ 1 week
we have two conversion factors: (7 days/week) and (1 week/7 days)
We use the first one because the unit required is days
3 weeks x ( 7days/week) = 21 days
Example 2. Let us consider the conversion of units problem : What length in centimeter corresponds to 2.00
inches?
The steps to solve this problem are :
1) Identify the given quantity: 2.00 inches
2) Identify the wanted quantity : ? centimeters
3) Find the conversion factor.
From the relation 2.54 cm ≡ 1 in. we can identify two factors:
(2.54 cm/1 in ) and (1 in/2.54 cm)
We use the first one because the unit required is cm.
4) Setup a unit path.
Quantity given  quantity to be found
2.00 in.  ? cm
5) Carry out the calculation by applying the rule mentioned previously ( always use conversion factor
that leads to the cancellation of an unwanted unit)
2.00 in. x 2.54 cm = 5.08 cm
1 in.
6) Check the answer to be sure that both the number and the units make sense.
The answer (5.08) is reasonable since we have large number ( 5.08 ) for smaller unit (cm) and small number
(2) for larger unit (in.). Should we use the second factor, we would have
2 in. x 1 in. = 0.816 in.2/cm
2.54cm
It is easy to recognize that this answer is wrong. If you get an answer with nonsense unit, you know you
have made a mistake.
4 SI UNITS
The International System was adopted for scientific use by the US National Bureau of Standards in 1964.
Prefixes The SI system is a decimal system, that is, one in which all derived units are multiple of ten
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Mass
Mass is a measure of quantity of mater. The metric unit of mass is gram, g.
The SI unit of mass is kilogram, kg. A penny has a mass of 3 grams.
Length
The SI unit of length is meter. A convenient unit for laboratory work is centimeter. 1 cm = 10-2 m . A
penny has a diameter of about 2 cm. One inch is defined as 2.54 cm
Volume
The volume of an object is the amount of space it occupies.
The SI volume unit is the cubic meter, m3. A more practice
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unit is the cubic centimeter, cm . It is the volume of a cube with edge of 1 cm. A common unit for
volume is the liter, L, which is defined as exactly 1000 cubic centimeters. A milliliter is exactly equal to 1
cm3 .
5 METRIC-USCS CONVERSIONS
The relationship between the metric and the English systems is given in table 3
Table 3 Metric and English Conversion Factors
Mass
Length
Volume
1 lb. = 453.6 g
1 oz = 28.3 g
2.20 lb. = 1 kg
1in. ≡ 2.54 cm, definition
1 ft = 30.5 cm
39.4 in.= 1 m
1.09 yd = 1m
1 mile = 1.61 km
1 qt = 0.946 L
1 gal = 3.785 L
1 in.3 = 16.39 cm3
1 ft3 = 2.832 x 104 cm3
Pressure
Energy
14.69 lb./in.2 = 1 atm ≡ 760 torr (definition)
1 calorie = 4.184 J
29.92 in. mercury = 1 atm ≡ 760 mm Hg(definition)
1 Btu* = 1.05 kJ
= 101.3 kPa
* BTU( British Thermal Unit) is the amount of heat required to raise the temperature of one pound of
water one degree Fahrenheit.
5 SIGNIFICANT FIGURES
A measurement always has some degree of uncertainty. The accuracy of a measurement is it closeness to
the true value. The precision of measurements indicates the degree of reproducibility of several
measurements of the same quantity. Accuracy is associated with systematic error, precision reflects the
random error.
Uncertainty of a measurement depends on many factors, one of which is the instrument used. The length of
a board is measured by comparing it with meter sticks that have different graduation marks. The result can
be 0.6 ± 0.1 m, 0.64 ± 0.01 m, 0.642 ± 0.001 m.
One way to express the uncertainty of a measurement is to use significant figures in reporting data. We use
the convention that the uncertainty in the last digit on the right is + 1. The number of significant figures is
the number of digits that are known accurately plus the first uncertain digit. Rules for significant digits are
presented in the following table
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Number
number of
significant digit
Rule
2.36 g
3
All nonzero digits are significant
6.087 g
4
Zeros between significant digits are significant
55.00 g
4
Zeros that end a number after the decimal are
significant
0.0295 g
3
Zeros that begin a number are not significant.
123000 g
3
Zeros that end a number before the decimal are
assumed not to be significant.
1.230x105 g
4
Exponential notation must be used for large
number to show if final zeros are significant
Significant figures do not apply to exact numbers. Numbers fixed by definition are exact. One inch is
exactly equal to 2.54 centimeter. Counting numbers are exact. Exact numbers have infinite significant
digits.
Rounding Off.
When experimentally measured quantities are added, subtracted, multiplied or divided, the result must
be rounded off to obtain the correct number of significant figures. Rules for rounding off are:
1) If the first digit to be dropped is less than 5, leave the digit before it unchanged.
2) If the digit to be dropped is 5 or greater, increase the digit before it by 1.
Significant Figures in Calculation
Rule for Addition and Subtraction
The number of decimal places in the result should be the same as the smallest number of decimal places
in the data.
12.4501 g + 2.36 g = 14.8101 g : round to 14.81 g
Rule for Multiplication and Division
The number of significant figures in the result should be the same as the smallest number of significant
figures in any factor.
1.234 x 0.025 = 0.03085 : round to 0.031
6 TEMPERATURE
The familiar temperature scale in the US is the Fahrenheit. Temperatures in chemistry are often
reported in Celsius scale. The SI temperature scale is the Kelvin scale ( also called absolute temperature
scale). The relationship between these scales are shown in table 4.
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Table 4. Temperature Scale
Reference
Temperature
Fahrenheit Scale
Celsius Scale
Kelvin Scale
Boiling point of
water
212 0F
100 0C
373 K
Freezing point of
water
320F
00C
273 K
Limit of lowest
temperature
-4590F
-2730C
0K
Conversion Formulas
TF = 1.8 TC + 32
TC = (TF - 32) x 100
180
T K = TC + 273
To convert from Celsius to
Fahrenheit multiply reading
by 1.8 and add 32 ( one
degree in Celsius
corresponds to 1.8 degrees
Fahrenheit )
To convert from Celsius to
Kelvin add 273
The Celsius and Fahrenheit temperature are the same at -40 0. At all temperature above -400, the
Fahrenheit temperature is the larger number.
DENSITY
Density of a substance is its mass per unit volume.
Density ≡ mass/ volume ;
D≡ m/V
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Substance with a high density ( Hg, Pb for example) have a much larger amount of matter in a given
volume than do substances with low density ( for example Al)
The specific gravity of a liquid is the ratio of its density over the density of water. Since the density of
water in SI units is 1.00 g/cm3, the specific gravity for a liquid is the same as its density without units.
Density of a substance may be used as a factor for conversions between volume and mass.
Strategy for Solving Problem
There are six steps:
1) List everything that is given, including the unit.
2) List all wanted quantities, including the unit.
3) Identify the relationship between the given and wanted quantities. Decide the method to solve the
problem. Algebra is a very convenient method if you know algebraic equations relating given and wanted
quantities. If given and wanted quantities are directly proportional to each other, dimensional analysis
method can be used. In using dimensional analysis, your verification that everything is correct is that you
end up with the correct unit. Some problems can be solved by both methods. Dimensional method has the
advantage of minimizing memorization.
4) Write the calculation setup for the problem. Include the unit.
5) Calculate the answer. Include the unit.
6) Check the answer. Be sure that the units are correct and the number is reasonable.
Example : The density of an oil is 0.862 g/mL. Find the volume occupied by 196 g of that oil.
Dimensional analysis method
Given :
mass of oil 196 g
Wanted
Volume of oil in ml
Path
g ( mass of oil)  mL ( volume of oil)
Factors
0.862 g/1 mL
and
1 mL / 0.862g
Set up and calculation
196 g x 1 mL/ 0.862 g = 227 mL
Algebraic method
Given :
mass of oil 196 g
Wanted
Volume of oil in mL
Equation
V = m / D = 196 g / (0.862g/mL) = 227 mL
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