CHEM 1411 Concept Reviews: Matter, Measurement, & Problem Solving Part II

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Significant Figures (Sig figs):

The numbers that reflect the degree of certainty to which a value is measured.

 

Measuring With Appropriate Significant Figures

    To measure a value to the appropriate number of significant figures, one must measure all digits known with certainty PLUS one estimated digit.  Digital instruments will do this for us, but for non-digital thermometers, meter sticks, graduated cylinders, etc., this means estimating between the smallest graduations (or markings) on the instrument.  (i.e.-If the smallest marking is worth 0.1, you must estimate to the nearest 0.01)

 

Identifying the number of Significant Figures in a Measured Value

    Whenever a numeric value is reported, it can be assumed that all non-zero numbers are significant.  However, zeroes are sometimes needed simply to hold place value but not to indicate the degree of certainty of the value.  When such zeroes are needed, we do not consider them significant.  The three simple Rules of Zero below can help you determine whether a zero is considered “significant”. 

 

The Rules of Zero (for Significant Figures)

1)  Leading zeroes (zeroes before non-zero numbers) are NEVER significant.          EXAMPLE:  0.00000259 m

2)  Sandwiched zeroes (zeroes between non-zero numbers) are ALWAYS significant.          EXAMPLE:  10029 J

3)  Trailing zeroes (zeroes after non-zero numbers) are ONLY significant IF the value has a decimal point.

                             EXAMPLE:  12500 lbs (NOT significant)                      12.500 lbs (significant)

 

Exact Numbers:  Numbers with no uncertainty that are treated as having unlimited significant figures.  These values                                         typically come from accurate counting of discrete objects (4 atoms or 3 chairs), defined relationships

                            (1 km = 1000 m), or from numbers as part of an equation (Circumference = 2πr).

 

 

Calculating with Significant Figures

    In order to ensure that our calculated values continue to express the appropriate level of certainty corresponding to the measurements used in calculation, rules are used to determine how many significant figures a calculated value should have.  They differ according to the mathematical process used in calculation.

 

Multiplication/Divison Rule:  The product or quotient of two or more measured values must have the same number of significant figures as the measured value with the fewest significant figures.

                                                          $3.210\:m\times6.5\:m=21\:m^2$

                                                                    4 S.F.             2 S.F.            2 S.F.

 

Addition/Subtraction Rule:  The sum or difference of two or more measured values must have the same error as the measured value with the greatest error.  The error of a measured value is in its estimated digit.

NOTE:  For addition/subtraction, it doesn’t matter how many sig figs there are, but rather where the LAST sig fig is.

EXAMPLE:  The estimated (uncertain) digit is bolded.

 

124.56             (The error is in the hundredths place)

+  4.5               (The error is in the tenths place.)

129.1               (Answer must have its final significant figure (estimated digit) in the tenths place.)

 

 

Accuracy:  The degree to which a measured value corresponds with the “true” value (sometimes called the “literature value”, as defined by scientific research literature and publications.)        

Precision:  The degree to which different measurements of a quantity or the quantitative result determined from different trials of an experiment agree with each other.  (In basic terms, precision is the consistency of your results.  The closer your measurements are to each other, the more precise you are.)

Random Error:  Uncontrollable errors (usually small) that have an equal probability of being too high or too low.  These can be treated with statistics.  With enough trials, they usually average out.

Systematic Error:  Error in technique or equipment that results in values being consistently too high or consistently too low.

 

 

Temperature Scales

The Celsius scale is based upon the freezing (0˚C) and boiling (100˚C) points of water divided into 100 equal increments.  The Kelvin scale is identical to the Celsius scale with regards to the size of increment but starts with its zero value (0 K) at absolute zero (the lowest temperature theoretically possible) instead of the freezing point of water.  The equation for changing between the two scales is as follows:

$K=\:^\circ C+273.15$

 

The Fahrenheit scale is based on the same two temperature reference points of the freezing and boiling points of water, though those reference points are given different values (32˚F for freezing and 212˚F for boiling).  Between the two points, there are exactly 180 increments, hence the use of the term “degrees”.  The equations for changing between the Celsius and Fahrenheit scales are as follows:

$^\circ C=\frac{5}{9}\left(^\circ F+32\:\right)$     Or     $^\circ F=\frac{9}{5}\left(^\circ C\right)+32$

 

 

Density Formula:

$Density=\frac{mass}{volume}$    or     $D=\frac{m}{V}$

 

 

Dimensional Analysis

A method of problem solving that keeps track of units and cancels them using conversion factors.

Conversion Factor:  A fraction where the numerator is equal (or at least directly related to) the denominator.  Any time you can set one value equal to another, those values can be used as a conversion factor.

EXAMPLES:  $1\:inch=2.54\:cm\:\:\:\Rightarrow\:\:\:\left(\frac{1\:inch}{2.54\:cm}\right)\:or\:\left(\frac{2.54\:cm}{1\:inch}\right)$

 

 

Dimensional Analysis Example Problem:  What is the velocity in m/s of a car driving at 65 mi/hr?

1)  Start the problem with what your given.  Write divided units like this:                        

 

2)  Whatever unit is initially in the numerator must come down for your next conversion factor.  This cancels out the unit.

 

3)  Continue the problem as follows:

$\frac{65\:mi}{1\:hr}\times\left(\frac{5280\:ft}{1\:mi}\right)\times\left(\frac{12\:in}{1\:ft}\right)\times\left(\frac{2.54\:cm}{1\:in}\right)\times\left(\frac{1\:m}{100\:cm}\right)\times\left(\frac{1\:hr}{60\:\min}\right)\times\left(\frac{1\:\min}{60\:\sec}\right)=29\:m/s$