Download Measurements, Significant Figures, Scientific Notation

Measurements, Significant
Figures, Scientific Notation
• "If you have only one watch, you always know
exactly what time it is. If you have two watches,
you are never quite sure..."
• The fact is, when we take a single reading, whether from a
thermometer, a wristwatch, or a fancy laboratory instrument, we
tend to accept the readout without really thinking about its validity.
People seem to do this even when they know the reading is
inaccurate. Your fancy digital watch is probably off by a minute or
two right now!
• Try as you might, with the most expensive instruments, under the
most ideal conditions, every measurement is subject to errors and
inaccuracies. But what is worse, modern digital instruments convey
such an aura of accuracy and reliability, that we forget this basic
rule...
• There is no such thing as a Perfect Measurement
Measurements
1. What is precision?
• The precision of an instrument reflects
the number of significant digits in a
reading or the closeness of a group of
measurements to each other (both are
related to how small of a measurement it
is capable of making and how consistent it
is)
Measurements
• The accuracy of an instrument reflects
how close the reading is to the 'true' value
measured.
Accuracy
•
When can you trust a measurement (is it
inaccurate)?
•
Consider when you would need to measure
temperature—you would have 3 options:
•
Glass Bulb Type - The expanding liquid is
constrained to a thin glass tube, and the height of
the liquid is a function of its temperature.
•
Dial Type - Working on the principle that different
metals expand at different rates when heated. A
metallic coil unwinds with heat and a pointer can
be attached to make the thermometer. If the
pointer is bent or forced beyond its normal range
of movement, inaccurate readings may result.
Accuracy
• Electronic Type - Working on the
principle that the electrical characteristics
of certain components (resistors, diodes,
transistors) varies with temperature, this
circuit essentially measures those
varying characteristics and converts the
result to a digital readout. Because the
characteristics do not vary uniformly, the
exact relationship between, (for
example) apparent resistance and
temperature, is not easily translated.
Outside of the operating range, very
unexpected results can occur and it may
not be obvious at all that the operating
range has been exceeded.
Accuracy vs. Precision
• Note that an accurate instrument is not necessarily
precise, and instruments are often precise but far from
accurate.
• For example, you might read out time right down to the
second, even though you know your watch is one minute
slow. This reading is precise, but not accurate.
• It makes little sense to quote values to high precision
beyond the expected accuracy of the measurement.
Without stating the estimated accuracy, such a reading
cannot be used in serious computations. Worse, even by
quoting the time down to the second, you have implied
some accuracy which you cannot justify.
Precision
• Any measurement must be recorded
in such a way as to show the degree
of precision to which it was made-- no
more, no less.
• Calculations based on the measured
quantities can have no more (or no
less) precision than the
measurements themselves.
• The answers to the calculations must
be recorded to the proper number of
significant figures. To do otherwise is
misleading and improper.
Significant Figures
• Determining the Number of Significant Figures
• The number of significant figures in a measurement,
such as 2.531, is equal to the number of digits that are
known with some degree of confidence (2, 5, and 3) plus
the last digit (1), which is an estimate or approximation.
As we improve the sensitivity of the equipment used to
make a measurement, the number of significant figures
increases.
• Postage Scale: 3
1g
1 significant figure
• Two-pan balance: 2.53 0.01 g
3 significant figs
• Analytical balance:2.531 0.001 g
4 significant figs
*Note that the accuracy is the same for all three scales, just
the precision is different
Significant Figures
• Rules for counting zeros as significant figures are
summarized below:
• Zeros within a number are always significant. Both 4308
and 40.05 contain four significant figures.
• Zeros that do nothing but set the decimal point are not
significant. Thus, 470,000 has two significant figures.
• Trailing zeros that aren't needed to hold the decimal
point are significant. For example, 4.00 has three
significant figures.
• If you are not sure whether a digit is significant, assume
that it isn't. For example, if the directions for an
experiment read: "Add the sample to 400 mL of water,"
assume the volume of water is known to one significant
figure. (Unless it’s written as 400. mL of water)
Significant Figures
• A simpler set of rules will also work:
1. Zeros that are found on the left of all
non-zero digits are never significant
2. Zeros that are found in between any nonzero digits are always significant
3. Zeros that are found on the right of all
non-zero digits are significant if there is a
decimal point present.
Significant Figures
• Practice determining the number of sig figs in
the following numbers:
• 2.3004
• 0.00004
• 0.00400
• 0.40040
• 230
• 230.
• 345,000
• 345,000.00
Calculating with Significant Figures
• Multiplication and Division With Significant
Figures
• The same principle governs the use of
significant figures in multiplication and division:
the final result can be no more precise than the
least precise measurement. So we count the
significant figures in each measurement:
• When measurements are multiplied or
divided, the answer can contain no more
significant figures than the least precise
measurement.
Multiplication and Division With
Significant Figures
• Practice multiplying or dividing the
following using appropriate sig fig rules:
• 230 x 12 = ?
• 0.4058 / 0.003 = ?
• 5482.3 / 25 = ?
• 74.077 x 2.100 x 16.0037 = ?
Calculating with Significant Figures
• Addition and Subtraction with Significant Figures
• When combining measurements with different degrees
of precision, the precision of the final answer can be no
greater than the least precise measurement. This
principle can be translated into a simple rule for addition
and subtraction: When measurements are added or
subtracted, the answer can include no more values
on the right side of the number than the least precise
measurement.
• 150.0 g H2O (using significant figures)
• + 0.507 g salt
• 150.5 g solution
Addition and Subtraction with
Significant Figures
• Practice adding or subtracting the
following using appropriate sig fig rules:
• 230 + 12 = ?
• 0.4058 – 0.003 = ?
• 5482.3 + 25 = ?
• 74,077 + 2,100 + 16,003.7 = ?
Rounding
• Rounding Off
• When the answer to a calculation contains too many
significant figures, it must be rounded off.
• There are 10 digits that can occur in the last decimal
place in a calculation. One way of rounding off involves
underestimating the answer for five of these digits (0, 1,
2, 3, and 4) and overestimating the answer for the other
five (5, 6, 7, 8, and 9). This approach to rounding off is
summarized as follows.
• If the digit is smaller than 5, drop this digit and leave the
remaining number unchanged. Thus, 1.684 becomes
1.68.
• If the digit is 5 or larger, drop this digit and add 1 to the
preceding digit. Thus, 1.247 becomes 1.25.
Scientific Notation
• To write a number in scientific notation:
• Put the decimal after the first digit and drop the
zeroes on the right.
•
• In the number 123,000,000,000 The coefficient
will be 1.23
• To find the exponent count the number of places
from the decimal to the end of the number.
Scientific Notation
• In 123,000,000,000 there are 11 places.
Therefore we write 123,000,000,000 as:
Scientific Notation
• In scientific notation, the digit term indicates the number
of significant figures in the number. The exponential term
only places the decimal point. As an example,
• 46600000 = 4.66 x 107
• This number only has 3 significant figures. The zeros are
not significant; they are only holding a place. As another
example,
• 0.00053 = 5.3 x 10-4
• This number has 2 significant figures. The zeros are only
place holders.
Scientific Notation
• Addition and Subtraction:
• All numbers are converted to the same
power of 10, and the digit terms are added
or subtracted.
• Example: (4.215 x 10-2) + (3.2 x 10-4) =
(4.215 x 10-2) + (0.032 x 10-2) = 4.247 x
10-2
• Example: (8.97 x 104) - (2.62 x 103) =
(8.97 x 104) - (0.262 x 104) = 8.71 x 104
Scientific Notation
• Multiplication:
• The digit terms are multiplied in the normal way
and the exponents are added. The end result is
changed so that there is only one nonzero digit
to the left of the decimal.
• Example: (3.4 x 106)(4.2 x 103) = (3.4)(4.2) x
10(6+3) = 14.28 x 109 = 1.4 x 1010
(to 2 significant figures)
• Example: (6.73 x 10-5)(2.91 x 102) = (6.73)(2.91)
x 10(-5+2) = 19.58 x 10-3 = 1.96 x 10-2
(to 3 significant figures)
Scientific Notation
• Division:
• The digit terms are divided in the normal way
and the exponents are subtracted. The quotient
is changed (if necessary) so that there is only
one nonzero digit to the left of the decimal.
• Example: (6.4 x 106)/(8.9 x 102) = (6.4)/(8.9) x
10(6-2) = 0.719 x 104 = 7.2 x 103
(to 2 significant figures)
• Example: (3.2 x 103)/(5.7 x 10-2) = (3.2)/(5.7) x
103-(-2) = 0.561 x 105 = 5.6 x 104
(to 2 significant figures)
Scientific Notation
• Powers of Exponentials:
• The digit term is raised to the indicated power
and the exponent is multiplied by the number
that indicates the power.
• Example: (2.4 x 104)3 = (2.4)3 x 10(4x3) = 13.824
x 1012 = 1.4 x 1013
(to 2 significant figures)
• Example: (6.53 x 10-3)2 = (6.53)2 x 10(-3)x2 =
42.64 x 10-6 = 4.26 x 10-5
(to 3 significant figures)