Read the following information and complete the questions that follow
UNCERTAINTY IN MEASUREMENT
Every measurement has an uncertainty to it. Even if you assume a measuring instrument is reliable, you still are estimating your reading in the last decimal point. To get a more accurate measurements, you can substitute an instrument with wider markings and obtain a better measurement, but the last place is still estimated. Even if you do not estimate the last place, it is because the measurement happened to fall precisely on a mark on the instrument.
In scientific work, the uncertainty in one's measurements is usually reported within the measurements themselves. When the uncertainty is specified, the reader can see just how reliable a particular measurement is. For example, if a chemist reports that in an experiment the temperature rose to 85 plus or minus 5 degrees Celsius, those results are not as precise as those of another chemist who reports a temperature of 84.5 plus or minus 0.5 degrees Celsius.Note that the uncertainty is cited after the measurement as figures to be added to and subtracted from the measurement. This notation indicates the range within which the measured value probably lies.
In deciding on the uncertainty of a measurement, you must take into account the closeness of the marks on the measuring instrument. If the marks are close together, than 0.5 of the smallest interval is probably the best you can do. On the other hand, if the marks are widely spaced, you usually estimate rather well to the nearest tenth of the interval.
Many modern instruments have "digital readouts," that is, the instrument reading is displayed electronically in numerals. The uncertainties in measurements made with these instruments are usually listed in the specifications section of the instruction book for the instrument. The uncertainty is usually different for various ranges of the instrument. Often, these uncertainty data are not immediately available in the laboratory. If no better information is available, you may assume an uncertainty of plus or minus 1 in the last significant digit of your measurement.
Many laboratory exercises result in the determination of a value that can be checked against accepted values in the scientific literature. For example, suppose a student has generated and measured a sample of oxygen gas. From her measurements, she has calculated the value of the molar mass of oxygen gas to be 29.92g/mol. The table of atomic masses is consulted, and the accepted value of the molar mass of oxygen gas is determined to be 32.00 g/mol. Aommon method of comparing two such values is the percent error, defined by
percent error = theoretical value - experimental value x 100%
theoretical value
Note that the numerator is an absolute value so that percent error is always positive. For the example above,
32.00 - 29.92 x 100% = 6.50%
32.00
EXAMPLE: By massing, heating to drive off water, and massing again, a student determines the percent of water in a hydrate of SrCl2 to be 40.8%. What is the percent error if the compound is actually SrCl2 .6H2O?
SOLVING PROCESS':
Actual percent of water in hydrate: 6H2O x
100% = 108 x 100% = 40.3%.
SrCl2 .6H2O
268
Percent error calculation: percent error= 40.3 - 40.8 x 100% = 1%
40.3
There are other situations in which you must use your judgement when making a measurement. Suppose you are timing a chemical reaction by starting a timer when the reagents are mixed and stopping the timer when a color change takes place. Both the start and stop points involve a judgement,. In such cases, you can improve the reliability of your measurement by repeating the same measurement several times. The probable time then lies somewhere within the range of times measured, most likely at or near the average time.
EXAMPLE: Determine the most likely time for a reaction from the following data. Assume that the reaction is complete when the color change occurs.
SOLVING PROCESS:
| Trial | Time t from mixing to color
change
|
| 1 | 1.8s |
| 2 | 1.8s |
| 3 | 1.5s |
| 4 | 1.7s |
| 5 | 1.7s |
| 6 | 1.6 |
total t = St = 10.1s
average t = 1.7s
The most likely time for this reaction is 1.7s.
In general, uncertainties should be reported to only one significant digit. That
digit should be the same decimal place as the last significant digit of the measurement.
For example,
Proper
Improper
36.5 ± 0.5 m
36.5± 0.25 m
300.4± 0.2 g
300.4± 0.05 g
280± 10s
282.4± 5s
The fractional uncertainty of a measurement is defined as the uncertainty divided by the measurement. Thus, if a mass is determined to be 28.1± 0.1 g, then the fractional uncertainty is
0.1g = 0.004
28.1g
If later, another mass is measured on the same balance as 81.8± 0.1g, the fractional
tional uncertainty is
0.1g = 0.001
81.8g
Now the wide range of fractional uncertainties obtainable on the same instrument with measurements of wide variation in magnitude. Fractional uncertainties are often converted to percentage uncertainties by multiplying them by 100%.
When using measurements in calculations, it is important to know how to combine uncertainties so that the final calculated value can be expressed with the proper uncertainty.
RULE 1: When adding or subtracting measurements, the uncertainty in the result is the sum of the uncertainties in the terms used.
36.8 ± 0.1 cm
19.1 ± 0.1 cm
+4.7 ± 0.1 cm
-2.5 ±0.1 cm
41.5 ± 0.2 cm
16.6± 0.2 cm
RULE 2: When multiplying or dividing measurements, the fractional uncertainty in the result is the sum of the fractional uncertainties in the factors used.
(41.7± 0.1 cm) (12.1 ± 0.1 cm)
fractional uncertainties:
0.1 cm = 0.002,
0.1 cm = 0.008
41.7
12.1
product: 41.4cm x 12.1 cm = 505cm
uncertainty:0.002 + 0.008 =0.01
0.01 x 505cm = 5cm
answer505 ± 5cm
Calculate the percent error in each of the following laboratory determinations.
a. The molar mass of CO2 is 43.79 g/mol.
b.The specific heat of Cd is 0.197 J/g.C. (The theoretical value is 0.231 J/g.C
c.The ionization constant of CH3COOH is 1.85 x 10-5. (The theoretical value is 1.75 x 10-5)
d.The melting point of tin is 244 C. (The theoretical value is 232 C)
2.The length of time for a separatory funnel to empty is measured as 6.04 ±0.03 seconds. What is the fractional uncertainty? What is the percentage uncertainty?
3.The temperature of a solution is measured as 100±3C. What is the fractional uncertainty? What is the percentage uncertainty?
4. A rectangular piece of aluminum metal measures 7.35±0.02 cm by 21.60±0.02 cm. Express the area of the piece with the proper uncertainty shown.
Activity 1
