Experiment 1: Measurements & Uncertainty



Experiment 1: Measurements & Calculations Lab

As discussed in class, the precision of the instruments with which we make measurements must be known. This part of the experiment examines that concept.

Part I: Determine the density [g/cm3] of a metal cylinder by two methods:

1. Use a graduated cylinder and beam balance;

2. Use the vernier caliper and analytical balance. Make sure you record the uncertainty of each measuring instrument. Calculate the absolute uncertainty and the percent uncertainty associated with each method.

Many times in chemistry, a particular dimension of a "piece" of matter is needed, but it is too small to measure directly. In those instances, one must approach the problem in an indirect manner. That is your task in this part of the experiment.

Part II: Determine the relative rate of reaction (qualitatively) of the zinc granules versus the iron filings with 4 M HCl. Use the spot plate provided.

Part III: Galvanized iron is made by placing a thin coating of zinc on the surface of iron. Obtain a sample of the galvanized iron. Using appropriate measuring devices, results from part II, and the theoretical density of Zn (7.13 g/cm3), determine the thickness [µm] indirectly, of the zinc on the iron. [While the uncertainty of all measurements is important, that is not the emphasis of this part of the experiment. Do NOT do those calculations for this part.] Enter onto the computer, the appropriate data so that the thickness of the zinc coating [µm] on one side of the galvanized iron can be determined. Make sure you take the mass of the galvanized iron before and after the reaction.

Summary: 1. State the absolute uncertainty and the % uncertainty of the density of the metal by each of the two methods. Discuss the relative uncertainty of each result. 2. In regard to the thickness of the zinc coating on the galvanized iron: Report your result and the mean and standard deviation of the class results. Discuss the individual and class precision by calculating the relative error for both. What assumption(s) must be made in this determination? Possible errors and effects?

Extra Equipment:

Spot plate

Vernier caliper

Analytical electronic balance

Triple beam balance

Graduated cylinder

Chemicals/Supplies:

Galvanized iron

Metal cylinders

Fe filings

Zn granules

Galvanized iron

4 M HCl [15 mL]

A simple method for Estimating Experimental Uncertainty[1]

To illustrate this method, we will consider the determination of the density of an irregularly shaped solid. In this determination, we make three measurements. First, we measure the mass of the object on the balance. Next, we must obtain the volume of the solid by water displacement. We can then calculate the density of the solid from the equation[pic], where M is the mass of the solid, V1 is the initial volume of the liquid in the graduated cylinder, and V2 is the volume of liquid plus solid. Suppose we get the following results: M = 23.06 g, V1 = 10.4 mL, V2 = 13.5 mL. The calculated density is[pic].

Now suppose that the precision of the balance used is ±0.02 g and that the volume measurements are precise to ±0.05 mL. How do we estimate the uncertainty of the density? We can do this by assuming a worst case. That is, we assume the largest uncertainties in all measurements, and see what combinations of measurements will give the largest and smallest possible results (the greatest range). Since the density is the mass divided by the volume, the largest value of the density will be that obtained using the largest possible mass and the smallest possible volume:

largest possible mass = 23.06 + 0.02 g

[pic]

smallest possible V2 = 13.5 – 0.05 mL - largest possible V1 = 10.4 + 0.05 mL

The smallest value of the density is:

smallest possible mass = 23.06 - 0.02 g

[pic]

largest possible V2 = 13.5 + 0.05 mL - smallest possible V1 = 10.4 - 0.05 mL

Thus, the calculated range is from 7.20 to 7.69 and the average of these values is 7.44. The error limit is the number that gives the high and low range values when added and subtracted from the average. Therefore we can express the density as 7.44 ± 0.25 g/mL, which is the average value plus or minus the quantity that gives the range calculated by assuming the largest uncertainties.

If we are to compare results which depend on different methods or quantities, we must “standardize” the uncertainty by transforming it to a percent uncertainty. In this case, the result would be transformed as follows: [pic].

Measurements & Calculations Lab: Determining the Uncertainty in Measurements

Example Calculations:

Initial volume of water = 15.0 ml [pic] 0.1 ml

Final volume of water with cylinder = 16.6 ml [pic] 0.1 ml

With Beam balance

Mass of Zn = 12.40 g [pic] 0.01 g

Max density = [pic]

= [pic]

= [pic] = 8.864 g/ml

Min density = [pic]

= [pic]

= [pic] = 6.883 g/ml

Range of density: 6.883 g/ml to 8.864 g/ml

Average density = [pic]

= [pic] = 7.873 g/ml

Absolute Uncertainty = 7.873 g/ml [pic] 0.99 g/ml

Percent Uncertainty = 7.873 g/ml [pic]([pic])

= 7.873 g/ml [pic] 12.57 %

With Analytical balance

Mass of Zn = 12.36 g [pic] 0.001 g

Diameter of cylinder = 1.27 cm [pic] 0.01 cm

Height of cylinder = 1.27 cm [pic] 0.01 cm

Max density = [pic]

= [pic]

= [pic]

= [pic]

= 7.867 g/cm3

Min density = [pic]

= [pic]

= [pic]

= [pic]

= 7.503 g/cm3

Range of density: 7.503 g/cm3 to 7.867 g/cm3

Average density = [pic]

= [pic] = 7.685 g/cm3

Absolute Uncertainty = 7.685 g/cm3 [pic] 0.182 g/cm3

Percent Uncertainty = 7.685 g/cm3 [pic]([pic])

= 7.685 g/cm3 [pic] 2.368 %

Summary

Density by pan balance & water displacement = 7.873 g/ml [pic] 12.57 %

Density by analytical balance & vernier caliper = 7.685 g/cm3 [pic] 2.368 %

Note that while the density of the metal cylinder is different by about 3%, the uncertainties of the densities vary considerably more. The uncertain of the measuring instruments themselves greatly affects the reliability of the calculated results.

[pic]

Simple Statistics

Comparisons Using Experimental Results

In your discussion of quantitative labs, you need to consider the precision and accuracy of your results. As indicated in your text, precision refers to the reproducibility of a measurement or a calculation based on measurements. Precision is normally expressed as a percent of closeness or agreement to an average experimental value. The complement of precision is relative error (RE). The sum of the two is 100%. Relative error can be viewed from two perspectives, (1) how your individual result compares to an average result: [pic]; or (2) the relative error of an entire group: [pic]. In either case, % Precision = [pic]

Accuracy refers to the agreement of a particular value to a true or theoretical value. Where RE is in correspondence with precision, we use absolute error (AE) in correspondence with accuracy. For an individual, [pic], while for a group, [pic], and then % Accuracy = [pic]

Definitions of Symbols:

[pic] where n = number of values in the experimental set

s = standard deviation of set of results [pic]

μo = known value (reference value)

RUBRIC: Experiment 1: Measurements & Calculations Lab

Name: ________________________

|Item |Points/Out of |

|Name, Partner, and Date Experiment Started |/2 |

|Title & # of Experiment |/2 |

| | |

|Appropriate Purpose: State the theory behind the experiment that is to be performed. What are the objectives? This should be 50 ± |/3 |

|5 words. Chemical equations should be used where possible. | |

| | |

|Materials & Chemicals: Tabular form or listing of all laboratory glassware, equipment, and chemicals to be used. Include the |/3 |

|amounts of all solid/liquid chemicals using metric units | |

| | |

|Clear & Complete Procedure: A listing of the steps that are to be taken in the experiment. It should be detailed enough so that |/3 |

|the experiment can be performed without the aid of the lab handout. | |

| |

|Clear & Complete Data and/or Observations: It is useful to display raw data and observations in the form of tables and then graph the results from the tables. |

|Each table should be titled at the top of tables. Each title is a complete sentence that provides enough explanatory information so that a reader can understand|

|all the information presented in the table or figure (e.g., Table 1. Relationship between enzyme activity and pH). Label table columns and graph axes completely|

|to prevent any ambiguity in interpretation. Graphs should be produced using a computer graphing program, printed out and taped into the lab notebook in the |

|appropriate section. Use balanced chemical equations where appropriate. |

| Part I |/2 |

| Part II |/2 |

| Part III |/2 |

| | |

|Calculations: The results section summarizes the raw data and observations found in your lab notebook. It should include all the calculations. For the |

|calculations; show all mathematical equations, work, and metric units whenever possible. |

|Clear & Complete Calculations for Part I |/10 |

|Clear & Complete Sample Calculations for Part III |/10 |

|Contribute Data to Class |/4 |

|Appropriate Analysis of Class Data & Statistics |/4 |

| | |

|Conclusion of Results & Discussion of Error: The reasoning from the observations/data to conclusions should be sound and logical: the conclusions should be |

|derived specifically from the data. Start by restating the purpose and hypothesis. Next, support, modify, or reject your original hypothesis. Then discuss what |

|you learned by summarizing your data/observations and results. Relate your observations to observations, concepts, and principles reported by other groups who |

|have conducted similar experiments. Do not discuss any results in this section that were not presented in the results section. Finally, comment on what went |

|wrong, by discussing all possible sources and effects of error. Also, discuss how you would change your laboratory technique to reduce error in future trials. A|

|percent error calculation should be reported whenever possible. Also, discuss any relative and absolute error. Sometimes comments are made about the direction |

|for further research. |

| Appropriate English Used (grammar, sentence structure, ect…) |/4 |

| Conclusion of Results |/2 |

| Discussion of Results |/5 |

| All possible sources of error |/5 |

| All possible effects of error |/5 |

| Signature |/2 |

| Date Experiment Handed In |/2 |

| | |

|Late (-5 until noon next day: no credit for later labs) | |

|TOTAL | /72 |

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[1] Adopted from Zumdahl, Steve. Chemistry. (2003) Houghton Mifflin Company

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Zn

Fe

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