Topic 11 – Measurement & Data Processing

name: _______________________

IB Spring 2015

Topic 11 – Measurement & Data Processing Background & Review

The Scientific Method

· Scientific Method – a logical process used to solve a problem

· The Scientific Method has five basic steps:

1) Define the problem(I wonder what causes that…?)

2) Form Hypothesis(I think it might be…)

3) Test Hypothesis (Experiment)(Let’s find out)

4) Collect and Analyze Data(Here’s what I found out)

5) Draw Conclusions(Here’s what it means)

· A hypothesis = a testable statement (this is a better definition than an “educated guess”).

· In an experiment, you get data, or information. This information can be one of two types: quantitative (numerical) or qualitative (non-numerical).

quantitative

qualitative

 

 

 

 

 

 

 

 

 

 

SI Measurement

· Fundamental units (standards) for SI:

 length:

  meter (m)

 mass:

  kilogram (kg)

 volume:

  liter (l)

 time:

  second (s)

 temperature:

  Kelvin (K)

· Conversions you should know:

· 2.54 cm = 1 inch

· 1.6 km = 1 mile

· 1 kg = 2.2 pounds

· 3.785 liters = 1 gallon

· 1 cm3 = 1 mL

· 1 dm3 = 1 L

· 0C = 273K

(but 1C increments = 1K increments)

· (KHD–DCM i e e e e i l c c c n l o t a i t l o i iKHD–DCM i e i r h i n n e i o l g r d n c k y k o i l n a g t e)SI prefixes:

Prefix

Symbol

Meaning

mega

M

1 000 000

kilo

k

1000

hecto

h

100

deca

da

10

----

----

---- 

deci

d

1/10

centi

c

1/100

milli

m

1/1 000

micro

1/1 000 000

Temperature

· Temperature:

· Celsius: set up with 0C = freezing pt. of water; 100C = boiling pt. of water

(in other words, it’s relative to the freezing and boiling points of water)

· Kelvin: 0 K = absolute zero (temp. at which all molecular motion ceases)

0C = 273 K

· So, to convert from C K: C + 273 = K

· temperature – the average kinetic energy of the particles in a substance

· We talk about temperature when we discuss

how hot or cold something is.

· heat – the total kinetic energy of the particles in a substance

· Heat flows from hot to cold.

· To distinguish:

· Imagine a bathtub filled with warm water and a cup filled with boiling water.

· The cup has a higher T.

· The bathtub has more heat (b/c there’s more water).

Uncertainty & Error in Measurement

Random Uncertainties and Systematic Errors

· Quantitative chemistry involves measurement. A measurement is a method by which some quantity or property of a substance is compared with a known standard. Chemists use many different instruments to make measurements, so it is sensible to question how reliable these measurements are when compared with the accepted standards.

If the instrument used to take the measurements has been calibrated wrongly, or if the person using the instrument consistently misreads it, then the measurements will always differ from the accepted value by the same amount. Such an error is known as a systematic error.

· An example of a systematic error would be always reading a pipette from the sides of the meniscus, rather than from the middle (bottom) of the meniscus (see figure to the right).

· Another example of systematic error would be from using a thermometer which had an air bubble or was originally miscalibrated in the factory, giving a consistently biased reading for the temperature.

(However, even when measurements are subject to systematic error, they may still yield accurate results when used to measure a difference, because the systematic errors cancel out. For example, consider the liquid in the graduated cylinders shown to the left. Whether a student a student measures the volume correctly (from the bottom of the meniscus) or incorrectly (from the sides of the meniscus), V will still equal 2.0 cm3.)

Likewise consider the scenario below in which two thermometers were used to measure the temperature increase in an exothermic reaction in aqueous solution. Even though the thermometers give different readings, they both give the same temperature difference, because any systematic errors will negate each other (assuming that the expansion of the liquid used in the thermometers is linear).

Another type of uncertainty that is less easy to control is the effect of random errors. Random uncertainties (random errors) occur if there is an equal probability of the reading being too high or too low from one measurement to the next. These might include variations in the volume of glassware due to temperature fluctuations or the decision on exactly when an indicator changes colour during an acid-base titration.

Because they are random, these errors can be reduced through repeated measurements (that’s why we always do multiple trials in labs), because there is an equal probability of the answers being too high or too low each time a reading is taken. However, repeated measurements will not reduce a systematic error, because they will always be either too high or too low because of the nature of the error.

Accuracy and Precision

· Accuracy refers to how close a measurement is to the standard value, or the literature or accepted value.

· A graduated cylinder used to measure exactly 25 mL of solution is likely to be much less accurate than a pipette used for the same purpose, because the pipette has been carefully calibrated to deliver exactly that volume.

· Precision refers to how closely several experimental measurements of the same quantity (made in the same way) are to each other.

· It is possible to have very precise readings which are inaccurate due to a systematic error. For example, all the students in a class may obtain the same, or very similar, results in a titration, but if the standard solution used in all the titrations had been prepared wrongly beforehand, the results would be inaccurate due to this systematic error.

· Think of it like a dartboard…

· A second example: Say you’re doing an experiment to test the freezing point of water. You

perform the experiment twice (three trials each time) and get readings of:

1) 36, 37, 36F

2) 31, 32, 32F

How would you describe the accuracy of these experiments? What about their precision?

· The accuracy of a particular measurement or calculation can be quantitatively determined by calculating the percent error or percent difference:

(% difference = x 100)

% error = x 100

I want you to use % error (the one without) absolute values in your calculations, because I want it to be clear whether your answers are too high or too low.

Significant Figures

Significant Figures in a Measurement

· The number of significant figures in a measurement are all the digits you can measure or read for certain, plus one more that you estimate.

· Consider the liquid in this graduated cylinder and in the thermometer…

Significant Figures in a Number

· To ascertain how many sig figs are in a given number, there are basically three rules.

First ask the question, “Is the # smaller or larger than 1?”

· If smaller than 1:

· Find the first non-zero digit. That digit, and every digit after it are significant.

· Every zero before it is NOT significant (they are just place holders).

· If larger than 1:

· If there is a decimal, then every digit is significant.

· If there is not a decimal, find the last non-zero digit. That digit and every digit before it is significant. Every zero after it is NOT significant (they are place holders).

Rounding

· We’ll just use the basic rounding rules you learned in elementary school. When rounding, look at the number just to the right of (after) the last significant digit.

· If it is less than 5, leave the last sig fig unchanged. (“round down” or “round off”)

· If it is 5 or higher, increase the last sig fig by one. (“round up”)

Significant Figures in a Calculation

· Addition & Subtraction with Sig Figs

· First line up the decimals of the numbers, then do the math (don’t round yet).

· Your final answer can only be as precise as the least precise number in the problem. Round off to same digit / decimal place as the least precise number in the problem (i.e. – the number ending furthest to the left).

(3 000 000 + 4 327 862 = ? 3 000 000 + 4 327 862 7 327 862 = ) (156.74 – 38 = ? 156.74 – 38 118.74 = )

· Multiplication & Division with Sig Figs

· Again, your answer can only be as precise as the least precise number in the problem. Your answer should have the same # of sig figs as the number in the problem with the fewest sig figs.

12.4 cm

x 3.2 cm =

Scientific Notation

· one digit to the left of the decimal

· coefficient should indicate the correct # of sig figs

· coefficient x 10exponent

· exponent = # of places you moved the decimal

· if you had to move the decimal to the right, the exponent is negative

(negative exponent = # smaller than one)

· if you had to move the decimal to the left, the exponent is positive

(positive exponent = # larger than one)

· Problems:

1. 74 000 4. 5.3 x 104

2. 0.000 050 5. 7.00 x 10–3

3. 0.000 602 6. 4.077 x 105

Uncertainties in Calculated Results

· When using measuring apparatus in the laboratory, always look to try to find the manufacturer’s statement regarding the degree of tolerance. If it is not present, then you will have to estimate the uncertainty.

· If you are making a single measurement with a piece of apparatus, the absolute uncertainty and the percentage uncertainty can be stated relatively easily. For example, consider measuring 25.0 cm3 of solution with a 25cm3 pipette that measures to 0.1 cm3. The absolute uncertainty is 0.1 cm3, and the percentage uncertainty can be calculated as:

% uncertainty = 100 100 = 0.4%

Consider a second example. Suppose you are performing a titration, in which you use a burette that claims to measure 50 cm3 with an uncertainty of 0.1 cm3. If you do actually use it to measure out 50.0 cm3, then the percent uncertainty is:

100 = 0.2%

However, we typically don’t use the entire volume of the burette during such procedures. If you only deliver 20.0 cm3 of solution from the burette during the titration, the absolute uncertainty is still 0.1 cm3, but the percentage uncertainty is now higher:

100 = 0.5%

· If you are adding or subtracting two volumes or masses (etc.), then the absolute uncertainties are added together to determine the absolute uncertainty of the calculated answer. For example, suppose we have already measured out (using the aforementioned burettes) two volumes of solutions into separate Erlenmeyer flasks, one of which measures 25.0 cm3 0.1 cm3, and the second measures 20.0 cm3 0.1 cm3. Then we add the two solutions together by pouring the solution from the first flask into the second flask.

· Given the uncertainty inherent to the burette for each measurement ( 0.1 cm3), the solution volume of the first flask could actually range anywhere between 24.9 cm3 and 25.1 cm3. Likewise, the actual volume from the second flask could actually range anywhere between 19.9 cm3 and 20.1 cm3. Thus, when you add the two solutions together, their actual volume could range anywhere between 44.8 cm3 and 45.2 cm3. So you record your final volume of the mixed solutions as: 45.0 cm3 0.2 cm3. Again, note that the absolute uncertainty of the final answer is the sum of the absolute uncertainties from the numbers you added or subtracted. In this case: 0.1 cm3 + 0.1 cm3 = 0.2 cm3.

· If you are multiplying, dividing, or dealing with powers, then percentage uncertainties should be used during the calculation, and then converted back into an absolute uncertainty when the final result is presented. This can be illustrated with results obtained from a titration used to determine the molar mass of an unknown acid.

· Let’s assume that 2.500 g of the unknown acid was dissolved in distilled water up to a volume of 250 cm3 in a volumetric flask, then added to a burette, from which it was dispensed during the titration. Let’s also assume that 25.0 cm3 of a standard solution of base was pipetted into an Erlenmeyer flask, in which the titration was performed.

· Four separate pieces of apparatus were used, each of which contributed to the overall uncertainty of the final calculation.

When using a balance that weighs to 0.001 g, the uncertainty in weighing 2.500 g of the unknown acid will be:

100 = 0.04%

Similarly, a pipette measures 25.00 cm3 0.04 cm3, giving a percentage uncertainty of:

100 = 0.16%

Assuming the percentage uncertainty due to the burette is 0.50%, and that of the volumetric flask is 0.40%, the overall percentage uncertainty is obtained by summing all of the individual percentage uncertainties:

Overall uncertainty = 0.04% + 0.16% + 0.50% + 0.40% = 1.10%

(There is no uncertainty for the Erlenmeyer flask since it was not used to measure out a solution volume, because it is not intended to do so – they are essentially just containers.)

So, if the molar mass of the acid is determined to be 129 g mol–1, then the uncertainty for this value is 1.10% of this value, or 1.42 g mol–1. The absolute uncertainty should always be expressed with a single significant digit. Therefore, the molar mass of the unknown acid should be given as: 129 1 g mol–1.

· If the literature value for the molar mass of the acid is 126 g mol–1, then the percent error for this experiment would be:

100 = 2.38%

If the % error is greater than the % uncertainty, then random errors alone cannot account for the difference between the accepted value and the experimentally determined value, and there must have been some systematic errors in the investigation (which you must identify and make suggestions to remedy). That is the case here, where the 2.38% error is greater than the 1.10% uncertainty. Clearly, there are other uncertainties here that we have not taken into account when calculating the 1.10% uncertainty. The standard solution of base was prepared beforehand by the teacher, so you have no way of knowing the uncertainty in its concentration, or its purity. There is also some uncertainty about exactly where the endpoint was taken during the titration (whether measured by pH meter or color change of an indicator). These additional uncertainties should be mentioned in the evaluation section of any lab report, even if they are not actually quantified.

If the % difference is less than the % uncertainty, then the % difference is not significant, and it can be explained simply by the random errors in reading the scale of the instruments. In other words, in these situations, there is no reason to doubt the veracity of the accepted, literature value for the value in question, and there is no indication that you made any substantial systematic errors while conducting the investigation.

You should also recognize when the uncertainty of one of the measurements is much greater than that of the others. This uncertainty will then have the major effect on the uncertainty of the final result, and the approximate uncertainty can be taken as being due to that quantity alone. This may be the case whenever a thermometer is used. Compared with an analytical balance, or accurate and precise volumetric apparatus, the percentage uncertainty in the temperature readings will be much greater, particularly when the temperature difference is relatively small. For example, let’s say that the temperature rise in a calorimetry experiment was 6.8C, and the thermometer you used to take this measurement had an uncertainty of 0.5C. Since you had to take two measurements (Ti and Tf), the absolute uncertainty of the T is 1.0C. This gives a % uncertainty of 14.7%, which is almost certainly much higher than any other uncertainties inherent in the experiment.

Graphical Techniques

· Graphs can be extremely useful in chemistry for presenting relationships (or a lack thereof) among data or calculated results. Graphs can visually demonstrate trends in a much more effective manner than can a simple list of numbers.

By fitting a line to the graphed data, when appropriate, we can use the graphs to predict unknown values. There are various methods to do so, including:

· Determining the x– or y–intercept

This is useful in situations such as the graph shown below, where you want to extrapolate a value at the axis. In this graph, you could extrapolate the volume of a gas at 0C, when you only took measurements between 20C and 60C.

· Measuring the slope of the line/curve

Finding the slope of a line or curve can be useful in several scenarios. For example, consider the graph below (of the concentration of some product, P, versus time). By measuring the slope of the curve at any time, t, during the experiment, you can determine the rate of the reaction at that particular point in time.

The following graphs are good examples of when it is appropriate to manipulate the data to produce a straight line graph. When investigating the relationship between pressure and volume for a fixed mass of gas, the first plot (of pressure versus volume) produces a curve. However, the second plot yields a straight line. Once this graph is in the form of y = mx + b, values for both the slope (m) and intercept (b) can be determined.

This is informative, because, according to the Ideal Gas Law equation (PV = nRT), the pressure of a fixed mass of an ideal gas is inversely proportional to its volume, at constant temperature. In other words, pressure multiplied by volume should always give a constant value (PV = k), which is the basis for Boyle’s Law.

Obviously, from the equation, “k” = “nRT”. This value (“k” or “nRT”) should be the slope of the straight-line graph.

· Extrapolation requires extending the graph to determine an unknown value which lies outside the range of the values measured. Below, right, is a graph of volume measurements for a fixed mass of a gas over a temperature range of 20C to 60C, taken in 10C intervals. The results were fairly linear, permitting the extrapolation of the volume of this gas sample at 80C.

· Interpolation involves determining an unknown value within the limits of the values already measured. Note that both graphs above are graphs of the same data set. Only the scales have been change, to allow for accurate extrapolation and interpolation. In the graph on the left, the units on the y-axis (volume, in cm3) have been expanded for greater resolution, to permit the interpolation of gas volume at 35C.

· The following points should be observed when drawing a graph:

· Plot the independent (manipulated) variable on the horizontal axis

· Plot the dependent (responding) variable on the vertical axis.

· Choose appropriate scales for the axes. (You don’t have to start at 0!)

· Use Système International (SI) units wherever possible.

· Label each axis and include the units.

· Once you’ve plotted all the data points, you must decide whether to simply “connect the dots” or draw the line of best fit. Occasionally, you do just “connect the dots”, such as graphs you will see in the Periodicity unit for melting points or first ionization energies of elements plotted against atomic number (# of protons). However, it is much more usual in chemistry to draw the line of best fit.

A line of best fit may be a straight line or a smooth curve. If you are trying to find the line of best fit, you need to take sufficient readings (data measurements). Two points are obviously guaranteed to produce a straight line, so that wouldn’t be valid. To be scientifically valid, a minimum of five readings is usually required.

· Give the graph a descriptive title.

12

½

½

½

-

½

Expected

Expected

Observed

Expected

Expected

Observed

-

3

cm

0.04

g

2.50

t

measuremen

y

uncertaint

absolute

3

3

cm

25.0

cm

0.1

3

3

cm

50.0

cm

0.1

3

3

cm

20.0

cm

0.1

g

2.500

g

0.001

3

3

cm

25.00

cm

0.04

g

126

g

126

g

129

-

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ø

ö

ç

è

æ

V

1

vs.

P