Regents Chemistry - Scarsdale Public Schools



Regents Chemistry

Measurements and Calculations

(Chapter 5 Notes)

[pic]

MEASUREMENT

SI System - “Le Systeme International d’Unites”

SI Base Units: (overhead)

Quantity Name Symbol

length meter m

mass kilogram kg

time second s

amount of substance mole mol 6.02 X 1023

temperature Kelvin K

SI Derived Units:

volume m3 (or L)

density kg/m3 (or g/mL)

concentration mol/m3 (or mol/L)

force newton N (m.kg/s2)

pressure pascal Pa (kg/ms2)

energy joule J (m2.kg/s2)

SI Prefixes:

Prefix Symbol Meaning Example

kilo k 1000X 1 m = .001 km or 1 km = 1000m

deci d 0.1X 1 d = .1 m or 1 m = 10 d

centi c 0.01X 1 cm = .01 m or 1 m = 100 cm

milli m 0.001X 1 mm = .001 m or 1 m = 1000 mm

micro u 1 X 10-6X 1 u = 1 X 10-6 m or 1 m = 1 X 106 u

nano n 1 X 10-9X 1 nm = 1 X 10-9m or 1 m = 1 X 109 nm

pico p 1 X 10-12X 1 pm = 1 X 10-12m or 1 m = 1 X 1012 pm

Note: for water,

1000g = 1 kg = 1 dm3 = 1 L (liter)

or, 1 g = 1 cm3 = 1 mL (milliliter)

Using Significant Figures in Measurements

What are significant figures?

Here we are not talking about models in bathing suits. We are talking about the proper way to record a measurement. In recording any measurement, the scale of the measuring instrument determines the number of sig figs. Your job is to understand how to correctly read a scale and to record the measurement with the proper number of sig figs!

Significant figures are the ones we are certain of plus one estimated one. The figures we are certain of are determined by the scale of our measuring instrument. If the scale, for example, has its farthest division to the one place, then we are certain of all digits to the one place and we estimate to the next place. Our estimate will then be in the tenth place and we would express this as + 0.1 or + 0.5 depending on how precisely we can estimate (i.e. can we visualize the space between the ones in tenths = + 0.1, or is the best we can visualize halfway between the ones = + 0.5).

Let’s take an example. Measure the length of the rectangles below using the rulers provided:

0 1 2 3 4

A

length = __________ + __________

0 1 2 3 4

B

length = __________ + __________

What is meant by the term precision?

Which of the above measurements is more precise? Why?

A student looked at the measurement in A and estimated it was one-quarter of the way past 2 and recording the length in measurement as 2.25? Why is this incorrect?

Problem: Record the length indicated using each of the rulers. Include the uncertainty. Which ruler is more precise? Why?

[pic]

Significant Figures Review

The accuracy of any measurement made in a chemistry laboratory will depend on the precision of the instrument used and the accuracy of the worker in using this instrument. Therefore, when reporting a numerical measurement which has a limit of accuracy, it is customary to retain one doubtful figure. For example, 7.68 is a number which contains three significant figures. The first two figures are certain, while the last one is uncertain. Unless otherwise given, the uncertainty is assumed to be to the 1 place. Significant figures are all those figures which are certain plus one uncertain one.

7.69 (max)

7.68 assume uncertainty = + 0.01

7.67 (min)

The Box-and-Dot Strategy for Counting Sig Figs

The box-and-dot method consists of three simple steps for determining the significant figures. The steps must be followed explicitly.

Step 1

Draw a box around all nonzero digits, beginning with the leftmost nonzero digit and ending with the rightmost nonzero digit in the number.

For example, drawing a box around nonzero digits in the number 0.0123012300 gives 0.0 1230123 00 Any zero(s) trapped or “sandwiched” between nonzero digits will necessarily be included in the box. For convenience, a digit or number surrounded by a box may be referred to as a “boxed” digit or “boxed” number, respectively.

Step 2

If a dot (decimal point) is present, draw a box around any trailing zeros.

Continuing with the above example, a dot is present in the expression 0.0123012300; therefore, trailing zeros are boxed, which gives 0.0 1230123 00

Step 2 uses the term dot in lieu of decimal point for reasons of brevity and ease of recall.

The position of a decimal point within a number is irrelevant – the only test for boxing trailing zeros is the mere presence of a dot.

Because the method steps are followed explicitly, it is understood that trailing zeros should not be boxed when a dot is not present. In other words, draw a box around trailing zeros if and only if a dot is present.

Step 3

Consider any and all boxed digits significant.

Boxed digits are significant, whereas digits that are not boxed are not significant. Continuing the example from Step 2, the expression 0.0 1230123 00 reveals nine digits surrounded by boxes. Therefore, there are nine significant figures. Specifically, the significant figures are: 1, 2, 3, 0, 1, 2, 3, 0, and 0.

Note that the box-and-dot method does not expressly address leading zeros. There is no need. Leading zeros, which are never significant, always lie to the left of the box drawn in Step 1 and are therefore excluded from consideration as significant figures by virtue of the explicitness of Steps 1 and 2. Those steps provide criteria for boxing (and thus rendering significant) trapped and trailing zeros only, to the exclusion of all other zeros.

In general, at least one, but not more than two, boxes will be associated with any given number. Step 1 always requires that a box be drawn. Step 2 only allows for a (second) box to be drawn when two conditions are met: (i) a decimal point is present, and (ii) one or more trailing zeros are present.

Let’s try some problems. How many sig figs are in each of the following numbers?

a) 123.01230?

b) 12301230?

c) 123.0123?

d) 10100?

e) 10000.0?

f) 0.00010?

The box-and-dot method also works for numbers written in exponential notation. In this case we ignore the “ten-to-the-n-term”. For example, how many sig figs are in Avogadro’s number, 6.022 X 1023?

An area of ambiguity involves numbers ending without a decimal point. How would you write the number five thousand so that it would have four significant figures? Writing the number as “5000” indicates the presence of one significant figure only. Writing the number as “5000.” does not follow the accepted convention. The ambiguity can be removed by expressing the number in scientific notation:

Getting back to the implied uncertainty in measured quantities and their min and max values, the following serve as an example:

1700 1601 1600.01

1600 1600. 1600.00

1500 1599 1599.00

We will see later that the smaller the differences between the max and min of a measurement, the more precise (or repeatable) the measurement is.

Numbers Rounded Off

In various calculations, the numerical answers can be rounded off to obtain the proper number of significant figures. When the number dropped is 5 or greater, the last significant figure is increased by one. For example, the number 5.166 is changed to 5.17, so that it contains three significant figures. When the number dropped is less than 5, the last significant figure remains unchanged. For example, the number 5.164 is changed to 5.16 so that it contains three significant figures.

When adding and subtracting the answer should be rounded off so as to contain the least accurately known figure as the final one.

Example, add 16.2

111.51

+ 4.853

132.563 = 132.6

The least accurate figure, in the example above, is the first decimal place so that the answer can have figures only to this place.

When multiplying or dividing, the answer should be rounded of so as to contain only as many significant figures as are contained in the least accurate number.

Example, multiply 4.123 x 5.12 = 21.10976 = 21.1 (3 sig figs)

Example, divide 6.1 by 3.16: 6.1/3.16 = 1.93 = 1.9 (2 sig figs)

Standard Notation for Numbers

The numbers obtained in some measurements or calculations may be extremely small or very large. In order to facilitate the writing of these numbers, they are expressed as powers of 10. For example, 1,000 can be written as 1 X 103. The exponent "3" means that the decimal place has been moved 3 places to the left. Another example is 175,000 which can be written 1.75 X 105. In a similar manner, a very small number such as 0.0000560 can be written 5.60 X 10-5. What is done is to move the decimal 5 places to the right with the exponent becoming negative. In any number so written, the digits before the power of 10 are considered significant figures. For example, the number 1.03 X 105 contains three significant figures.

Mathematical Calculations with Exponent Numbers

Employ the following rules when using exponential numbers in mathematical operations:

1. If the original number contains a definite number of significant figures, the calculated number shall also contain the same number of significant figures.

2. Numbers expressed in powers of 10 cannot be added or subtracted directly unless the powers of 10 are the same. If they are not the same, the numbers can be rewritten so that the powers of the numbers are the same.

Example, 2.76 X 105 + 2.54 X 106 must be rewritten so both numbers have the same exponent:

= 2.76 X 105 + 25.4 X 105 = 28.2 X 105

Since the original number contained 3 significant figures, the final calculated value must also contain 3 significant figures.

3. When numbers with the power of 10 are multiplied, the exponents are added; when divided, the exponents are subtracted.

Example:

(2.76 X 105) X (2.54 X 106) = (2.76 X 2.54) X (105 + 106) = 7.01 X 1011

Example:

2.76 X 105/2.54 X 106 = (2.76/2.54) X (105/106) = 1.09 X 10-1

4. When obtaining the square root of numbers with powers of 10, the exponent is divided by 2; when obtaining the cube root, the exponent is divided by 3.

Example:

The square root of 16 X 108 is 4 X 104.

The cube root of 8 X 109 is 2 X 103.

Dimensional Analysis - Factor Label Method

Almost all the calculations we will be doing in chemistry involve using proportions to convert from one unit/quantity to another unit/quantity. One proven method for doing this is called the “factor label” method and it involves a fraction showing an equality between two units. This fraction is called a conversion factor. For example, in converting kilograms to grams, the equality is

1 kg = 1000g

and the conversion factor is: [pic] or [pic]

Let’s try a simple one step conversion problem:

How many grams are in a 1.50 kilogram sample?

Step 1: write the given quantity and unit:

1.50 kg

Step 2: multiply the given by the conversion factor gives the unit converting to in the numerator (top) and the unit converting from in the denominator (bottom):

1.50 kg X [pic]

Cancel units (dimensional analysis) and do the math:

1.50 kg X [pic]

The equality 1 kg = 1000g is known as a defined equality and does not involve measured numbers. These are called exact numbers and they do not have a limited number of significant figures. Therefore, these exact numbers (defined equalities) are not considered in our calculation of significant figures.

Problem: In France gasoline is sold by the liter. If you wanted 10.0 gallons of gasoline, how many liters should you ask for? (1 gal = 3.78 L)? How much would this cost in dollars?

Let’s try a problem involving converting two units.

Problem: The national speed limit is 55.0 mi/hr. What is the speed limit on km/hr? (1 mi = 1.6 km)

Let’s try a multistep conversion problem.

Problem: The Sun is 92, 000, 000 miles from Earth.

a) How far away is this in centimeters? (use scientific notation)

b) Given that light travels 3.00 X 1010 cm/s, how many seconds foes it take for light to travel from the Sun to the Earth?

Random and Systematic Error

Precision and Random Errors

Precision refers to how close repeated determinations done the same way are to each other. We like to be as precise as possible, but this does not always insure accuracy. We assume that the error between repeated measurements is due to random errors. Random error (or indeterminate error) means that a measurement has an equal probability of being low or high. This error occurs when estimating the value of the last digit of a measurement.

Let’s consider a series of measurements given below:

Trial Measurement

1 4.93

2 4.83

3 4.98

4 4.88

5 4.96

6 4.90

What is the best estimate for the value of these measurements? Calculate the average value:

Average = ____________

The average though, doesn't tell us anything about the uncertainty. The average of one hundred measurements should have less uncertainty than the average of five measurements. Therefore, a measure of uncertainty should take into account both the spread of the data and the number of determinations. This method is called the standard deviation and is given by s:

[pic] [pic]

where s = standard deviation from the mean, Xi = members of the set, X = mean, and N = number of members in the set of data. The symbol ( means to sum over the members; the symbol [ ] means absolute value, so all the differences are positive.

One way to make the calculation of standard deviation less cumbersome is to set up a data table like the one below:

Trial Measurement Deviation from X (Deviation)2

(Xi - X) (Xi-X)2

1 4.93 __________ __________

2 4.83 __________ __________

3 4.98 __________ __________

4 4.88 __________ __________

5 4.96 __________ __________

6 4.90 __________ __________

Average = __________ Sum of Deviation2 = _______

Standard Deviation(s) =

A practical application of standard deviation is that if any individual measurement deviates by more than two standard deviations from the mean (average), than that result is not statistically valid and should be discarded. This is known as the 2s rule.

Random error is only one source of error. The other source of error is called systematic (or determinate) error. Systematic error causes a shift from the true value and reduces the accuracy of the result. By making more measurements we can reduce the random error and increase the precision of our results, but if systematic errors exist the experimental result will still differ from the true (or accepted) value. Systematic errors can result from miscalibration of the instruments or some error in the experimental technique.

Uncertainty in Measurements

Some definitions:

absolute uncertainty (au) - the uncertainty in the measurement (i.e. the + )

relative uncertainty (ru) - (the absolute uncertainty/the measurement) X 100

When adding or subtracting measured quantities, the uncertainty of the sum or difference is determined by adding the absolute uncertainties of each measurement.

Example: 26.60 + .01 cm

- 20.00 + .01 cm

6.60 + .02 cm

When multiplying or dividing measured quantities, the uncertainty in the product or quotient is determined by adding the relative uncertainties of each measurement.

Example 12.42 + .01 X 6.35 + .05 = 78.86700 = 78.9 + ?

ru in 12.42 = [pic] X 100 = .08%

ru in 6.35 = [pic] X 100 = .8%

.08 + .8 = .88 = .9%

therefore, final answer = 78.9 cm + .9%

and .9% of 78.9 cm = 0.7 cm

so the answer can also be expressed as 78.9 cm + .7 cm, which is a way of expressing the maximum and minimum possible values discussed below.

Min & Max Calculations

Another way to express uncertainty in manipulating measured quantities is to calculate a minimum value and a maximum value based on the absolute uncertainties of the measurements. The average of the min and max is thus the final answer and the uncertainty is how much the average differs from the min or the max (the difference should be the same).

Example: Calculate the density of an object having the following mass and volume:

mass: 24.62 g + 0.01g

volume: 12.7 mL + .1 mL

Min density = [pic] Max density = [pic]

= [pic] = [pic]

= 1.92 g/mL = 1.96 g/mL

AVG = 1.94 g/mL + 0.02 g/mL = final answer

Measuring and Calculating the Perimeter of the Index Card

Uncertainty in ruler: + _____ cm

l = __________ + _____

l = __________ + _____

h = _________ + _____

h = _________ + _____

perimeter = _______________ + __________

Now do a Min and Max calculation:

Min = __________ Max = __________

__________ __________

__________ __________

__________ __________

( =__________ ( = __________

AVG (average) = [pic], and the + is the difference between the average and the min and the max)

AVG = _______________ + __________

Calculating the Area of the Index Card

l = __________ + _____ calculation for ru in length measurement:

(from above)

ru = __________ %

h = __________ + _____ calculation for ru in height measurement:

(from above)

ru = __________ %

area = __________ + _____ %

Now do Min and Max for the area calculation:

min length: __________ max length: __________

min height: __________ max height: __________

min area: __________ max area: __________

AVG = __________ + _____

How does the + from the min/max calculation compare to the relative uncertainty?

Error in Measurements – A Summary

Random Error Systematic Error

this is the + this is error that

(or absolute uncertainty) consistently made in

the same direction

equal chance of being

high (+) or low (-) may be due to a mis-

calibrated instrument

associated with precision -

how close measurements are associated with accuracy -

to each other (i.e. reproducibility) how close your measurement

is to an accepted value

we measure precision by

calculating standard deviation (s) - we measure accuracy by

this takes into account both the calculating percent error -

number of measurements and the

spread (high-low) of the data % error = obs - acc X 100

acc

[pic] [pic]

where Xi = individual measurement

X = average

N = number of measurements

Temperature

• a measure of the average kinetic energy of the atoms/molecules in a body

• the property or condition of a body that determines the direction of heat flow between it and another body (law: heat flows from hot to cold until thermal equilibrium)

How do we measure temperature?

All thermometers have two fixed points:

[pic]

How do we convert from one temperature scale to another?

A couple of density problems:

1. Given the following data:

Mass 21.58 g + 0.01 g

Volume water in cylinder 10.0 mL + 0.5 mL

Volume water and object 21.5 mL + 0.5 mL

Calculate the density of the object and include the uncertainty.

2. Given the following data:

Mass 9.55 g + 0.01 g

Length 2.00 cm + 0.01 cm

Width 1.00 cm + 0.01 cm

Height 3.60 cm + 0.01 cm

Calculate the density and include the uncertainty.

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