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experiment 3
LECTURE AND LAB SKILLS EMPHASIZED
- Determining what information is needed to answer given questions.
- Developing a procedure which allows you to acquire the needed information.
- Taking quantitative measurement on a sample.
- Learning how to use basic laboratory equipment.
- Using quantitative data to determine density mathematically and graphically.
IN THE LAB
- Students will work in pairs.
- Parts may be completed in any order.
- Record your procedure and original data in your lab notebook along with your calculations.
Introduction to Data Analysis
INTRODUCTION TO DATA ANALYSIS
- Report data collected and subsequent calculations to www.chem21labs.com.
- All equipment should be returned to the correct location after use.
WASTE
- Aluminum foil can be disposed of in the trash can.
- If you use any liquids other than water, contact your TA for disposal information.
SAFETY
- Safety goggles are mandatory when anyone is performing an experiment in the lab.
- Long pants, closed-toed shoes, and shirts with sleeves. Clothing is expected to reduce the exposure of bare skin to potential chemical splashes.
- Aprons are available for students wishing to have an extra layer of personal protection.
- Even if you are not working, you cannot anticipate what others might do. This rule is understood to be effective for the remainder of the semester, unless otherwise noted, and will not be repeated for each experiment.
Additional information can be found at http://genchemlab.wordpress.com/3-mass-volume-and-units/
As your first project at the research facility, your supervisor wants you to demonstrate that you have successfully learned the skills demonstrated in your training session and can apply what you’ve learned in school to answer two questions: 1) what is the density of aluminum foil, and 2) are aluminum foil sheets a cost-effective approach to their current source of aluminum? You must de- velop two methods or procedures to determine the density by thinking about what you can mea- sure in the lab and what data you need to answer the questions your supervisor asks for in the data analysis section. You must have this procedure to follow in your lab notebook prior to completing the experiment in order to obtain the necessary data to complete the experiment. Here are some suggested things to think about when developing your procedure:
- How many times should you measure the mass of the aluminum?
- Should you measure samples of different sizes?
- What factors should be considered when determining the volume of aluminum by displacement?
- What shape is best for determining the vol- ume by displacement?
- Will aluminum sink or float in water?
- What equipment is available to you in the lab?
Some information that you should know:
- A box of foil sheets costs $9.28.
- The box contained 500 sheets of 12" 10¾" foil. (This is not the size you will get.)
- You will need at least five data points (mass– volume pairs) to complete the data analysis. You should collect six or more.
- Dirty glassware has an effect on every experi- ment you do.
INTRODUCTION TO DATA ANALYSIS
to be wrong by more than two units. In the labora- tory, always record the proper number of signifi- cant figures for the measuring device you are us- ing. You should record neither less nor more than this number of figures.
Treating Measurements in
Calculations
Your measurements are treated in calculations by the same rules for determining significant figures as you were taught in lecture.
- Nonzero digits in a reported value are always significant.
Example: The number 5.4 has two signifi- cant figures, but the value of 5. has three significant figures.
- Zeros are not significant when they are after a decimal point but before the first nonzero digit. If the zero comes after a decimal point and after a nonzero digit, it is considered significant.
Example: In the value of 0.003, only the 3 is significant, and the number has 1 significant figure, but the value of 0.0030 has two significant figures.
- Zeros between nonzero digits are significant.
Example: The value of 5.04 has three signifi- cant figures, but the value of 5. has four significant figures and the value of 5.0404 has five significant figures.
- All digits in scientific notation are significant.
Example: The number of significant figures in the value of 400 is ambiguous. By writing the number in scientific notation, i.e., 4.00 103 , we can eliminate that ambiguity and state that it has 3 significant figures.
- Exact numbers obtained by counting or from definitions are exact numbers.
Example: To calculate the average of the values 115, 125, and 139, we sum them up and then divide the sum by 3. Since the number 3 is an ex- act number, the result is reported to three significant figures based on the numbers being averaged. Most exact numbers are whole numbers (i.e., there are 5 people on the elevator); that is not always the case. One inch equals exactly 2.54 cm.
Addition and Subtraction
When adding and subtracting numbers, retain only as many significant figures in the decimal portion of the number as in the least significant of the values. The nonzero digits to the left of the decimal are all significant.
Example: 76.32 5.465 0.58543 82.
The sum of the three numbers above can have only two significant figures after the decimal point, be- cause the number 76.32 is the number with the least number of significant figures after the deci- mal point (only 2).
Multiplication and Division
When multiplying and dividing numbers, retain in the result only as many significant figures as are contained in the number with the least number of significant figures.
Example: 0.286 25.44 3.6453 26.
The product of these three numbers can have only three significant figures, because the least accu- rately known number (0.286) has only three sig- nificant figures.
INTRODUCTION TO DATA ANALYSIS
Rounding
In rounding off numbers (the number of signifi- cant figures is reduced), the last digit retained is increased by 1 only if the following digit is a 5 or greater.
Examples:
If 6.5457 is to be rounded to 4 significant fig- ures, it would become 6.546.
If 6.5453 is to be rounded to 4 significant fig- ures, it would become 6.545.
PART II—CONSIDERATIONS TO
UNCERTAINIY
How certain are you with your measurement? Would someone else measure the volume of wa- ter in a graduated cylinder differently? How can you express to someone else your uncertainty with your measurement? To answer these ques- tions, scientists consider accuracy and precision of their measurements and quantify the accuracy and precision using mathematical techniques such as standard deviation and percent error (among other methods).
Precision and Accuracy
Many of the experiments you will perform in this course require you to measure various physical quantities, such as mass, volume, or temperature, and to use those data to calculate the value of an
unknown quantity. In order to obtain good results, you must know proper techniques for collecting data, and recognize possible errors in the process. You must also be able to evaluate the quality of the data you collect, and to state your results in a meaningful way. Since an experimentally deter- mined quantity always has some associated uncer- tainty, it is important to be able to give an estimate of this uncertainty in your results.
In the process of evaluating your data, you will:
- determine the precision and accuracy of your results.
- subject questionable data points to data rejec- tion methods.
- calculate the percent error in your answers.
Precision refers to the agreement among a set of measurements that were made in exactly the same way. There are several ways to express the degree of precision. One is as “deviation from the mean”; another is as “relative deviation.” Methods for cal- culating these are discussed in the next section.
Accuracy refers to the agreement between a mea- sured value and its accepted, or “true,” value. It is usually expressed as error, either absolute or per- cent. Calculation methods are discussed in the next section.
Figure 3.1 illustrates the difference between preci- sion and accuracy.
High accuracy High precision
Low accuracy High precision
High accuracy Low precision
Low accuracy Low precision
©Ha yden
il, LL C
Figure 3.1.
INTRODUCTION TO DATA ANALYSIS
Step 3: Calculate the absolute value of the devia- tion of each result (d | Xi X |), the sum of the deviations (Σ | d | ), the square of d values ( | d |^2 ), and the sum of the square of the d values (Σ | d |^2 ). Tabulate these values in a new table:
Experiment Number Xi | Xi X | | Xi X |^2
1 2.60 | 2.60 – 2.720 | = 0.12 0. 2 2.90 | 2.90 – 2.720 | = 0.18 0. 3 2.70 | 2.70 – 2.720 | = 0.020 0. 4 2.90 | 2.90 – 2.720 | = 0.18 0. 5 2.50 | 2.50 – 2.720 | = 0.22 0.
n 5 Σ Xi 13.60 Σ (Xi X)^2 0.
Step 4: The standard deviation can then be calcu- lated from the formula:
s 5 1
= 0.13^ 0.
^ h
Thus, we could state that the result of the density determination together with its standard deviation is 2.72 ± 0.18 g/mL. Note that the average cannot have more significant figures than the measure- ments that make up the average and that the stan- dard deviation has the same number of decimal places as the average.
As the value of the standard deviation tells us about the variation or spread of our data points within our measurements and its uncertainty, it does not make sense to include more than one significant figure in the standard deviation. Our measure- ment is correctly reported as 2.7 ± 0.2 g/mL. This indicates that our measurements were variable or not precise.
It is very important to realize at this stage that you can have a very small deviation in your data (indicating high precision) but your result may be significantly off from the true value (if the ac- curacy is low).
Percent Error
When doing an experiment that has been done be- fore, it is useful to evaluate the quality of the results because it gives you an indication of how well you did the experiment. Percent error can be used to determine the accuracy of the results. Accuracy is how close your experimental value is to the accept- ed value. Precision is how close your values are to one another for multiple trials of the same experi- ment. One way to evaluate the accuracy of your results is to determine the percent error in your experimental value using the equation shown.
% error accepted value
accepted value experimental value = #100%
Note the absolute value sign in the formula which results in the percent error always being a positive value. While having a low percent error is impor- tant, in this course, the focus will be on learning how to calculate the percent error and understand- ing why, if you have a high percent error. Regardless of the value of the percent error, you should always include the calculation in your results section and include it in the summary of results presented in your discussion. Additionally, you should also dis- cuss potential sources of error in the discussion even if you have a low value for the percent error. A low percent error doesn’t necessarily mean that everything was done perfectly, since you could have had two sources of error that offset one an- other. One way to come up with potential sources of error is to look at each step of the procedure and ask yourself, “What could have gone wrong in this step?” See the information on lab reports for more detailed information about what to include in your discussion including information about sources of error.
INTRODUCTION TO DATA ANALYSIS
Example Refer to our previous standard deviation calcula- tion. We determined the density of our liquid to be 2.72 g/mL. Suppose that the accepted value in the literature for the density is 2.43 g/mL. The relative or % error of the result is:
.
..
% error accepted value
accepted value experimental value 100%
100% 2 43
2 43 2 72
=
=
The precision of the experiment is given by the average value ± standard deviation , while the ac- curacy of the experiment is given by the percent error. In our case:
precision: 2.72 ± 0.
accuracy: 11.9% error
It is important to note that you cannot comment on the accuracy of the experiment unless you know the actual value of the unknown that you are investigating.
PART III—DENSITY
CONSIDERATIONS
Density (d) is a mathematical combination of mass (m) and volume (V) which all states of matter have as shown in the following equation.
d V =m
While there can be a variety of units for density, it is typically reported as g/mL for most solutions and solids.
Density is an example of a physical property of a substance and can be used to identify an unknown because the density of a sample at a given tempera- ture is constant. It is important to note that the density is constant at a given temperature because as the temperature changes, so will the density. Look in Table 3.1 to see how the density of water changes with temperature. Why does the density of water change when it freezes? Does the mass of the sample change with temperature? Does the volume of the sample change with temperature? The mass of the water stays the same, but the vol- ume increases as the water temperature decreases which leads to a lower value of density. Water is fairly unique in this respect since most substances experience a decrease in volume with a decrease in temperature.
Table 3.1. Density of water as a function of temperature.^4
Temperature (°C) Density (g/mL)
0 0. 20 0. 40 0. 60 0. 80 0. 100 0.
Many of the solutions used in this course will be aqueous (water-based) solutions. Water has a den- sity of 1.00 g/mL at 20°C. Substances that have a density greater than 1.0 g/mL will sink when placed in water while substances with a density less than 1.0 g/mL will float. Ice floats because it has a density of 0.98 g/mL.
4 “Properties of Water in the Range 0–100°C” in CRC Handbook of Chemistry and Physics, ed. David R. Lide, 6–10. Boca Raton: CRC Press, 1993.
INTRODUCTION TO DATA ANALYSIS
Data Analysis
Make sure to show all of your calculations in your lab notebook as a record of how you completed your calculations. Don’t forget to include your units and correct number of significant figures! Then, go onto Chem21 and report your results.
- Determine the cost per square inch of the alu- minum foil. Report answer in units of dollar/ in. 2
- Determine the grams per square inch using the data in Trial 1.
- Determine the cost per gram for the alumi- num foil. Report answer in dollar/grams.
4–8.Determine the density of the aluminum foil for each trial (1–5).
- Determine the average density of the alumi- num foil. 10. Determine the standard deviation for the den- sity of the aluminum foil. 11. Determine the density of the aluminum foil using the graphical method. 12. Using an online or print source, find the ac- cepted value for the density of aluminum, including the temperature at which is it accu- rate. Include a reference to the source in your lab report. 13. Determine the percent error in the density based on your average calculated density val- ue. 14. Determine the percent error in the density based on your graphical density value.
