Making Measurements
Introduction
All measurements involve two types of error, precision and accuracy. The precision error shows the reproducibility of
the measurement. An object is masses by four students as: 9.8g, 10.2g, 10.0g, and 10.4g. The measurements are averaged:
9.8g+10.2g+10.0g+10.4g =10.1g average mass
4
Each individual error is then calculated as the absolute difference between the measurement and the average.
The precision error for the experiment is equal to the average of those errors:
Student 1 |10.1 - 09.8 g| = 0.3 g error
Student 2 |10.1 - 10.2 g| = 0.1 g error
Student 3 |10.1 - 10.0 g| = 0.1 g error
Student 4 |10.1 - 10.4 g| = 0.3 g error
Total error
= .08 g
Average (Precision Error) = 0.8 g/4 = ± 0.2 g
The measurement is reported as 10.1 ± 0.2 g. The precision error is a guide to the proper use of an instrument. If the error
is large compared to the size of the measurement a more sensitive instrument should be used. For example, it would be
meaningless to weigh objects of 0.1 g or less on a balance whose average error is ±0.2 g. The triple beam balance whose
average error is about ±0.01 g would be a better choice. In most laboratory work, precision errors are taken into account
by recording all data using significant figures. For the triple beam balance all measurements should be recorded to the
nearest 0.01 g.
The accuracy error shows the deviation of a measured value from the true value. Measurements will differ from the true
value because of errors in the instrument, the particular technique being used, or to some quirk of the experimenter. The
accuracy error is calculated by using the following equation:
% Accuracy Error = |True Value - Measured Value| x 100
True Value
To measure the density of a substance, you measure the mass and volume of a sample of that substance. If the substance
is a liquid or an irregular solid, you can determine its volume with a graduated cylinder. If the substance is a solid with a
regular shape, you can obtain its volume by direct measurement of its dimensions. To find the density divide the mass of
the sample by the volume of the sample.
The thickness of a foil is difficult to measure directly. We measure it indirectly by measuring the mass of the foil and
using its density to calculate its volume: Volume = Mass/Density. area of the foil.. The thickness can then be found by the
following equation:
Thickness (cm) = Volume (cm3)/ Area (cm2)
Calculate the diameter of an atom of aluminum in nanometers and how many atoms are stacked high to equal the
thickness.
d=2r
thickness (nm)/diameter of the atom
Table 1 The Densities of Some Common Metals
Substance
Density (g/mL)
Aluminum
2.70
Germanium
5.32
Zinc
7.13
Chromium
7.20
Substance
Iron
Copper
Bismuth
Lead
Purpose
1. To indirectly determine the thickness of a metal foil.
2. To determine how many atoms of aluminum are stacked up to equal the thickness.
3. To identify an unknown metal by determining its density.
4. To determine the precision and accuracy of a measurement
Density (g/mL)
7.86
8.92
9.80
11.35
Procedure
1. Obtain a rectangular piece of metal foil from your instructor. You will be told the identity of the foil. Look up its
density on Table 1. Measure the mass, length, and width of the foil as precisely as your equipment will allow. Record your
measurements using significant figures.
2. Obtain a sample of an unknown metal. Weigh the container and its contents. Partially fill a 50 ml graduated cylinder
with water. Measure and record the water level. Carefully slide the substance into the water in the graduate and record the
new volume. The water must completely cover the substance. Now weigh the empty container. Dry the sample and return
it.
3. Mass the class object on your balance and record its mass. All students should weigh the same object. All the
measurements should be recorded on the blackboard and in your data.
Data
Thickness of a foil Density of a Substance
Radius of an aluminum atom
Metal
Diameter of an aluminum atom
Density
Mass
Length
Width
Unknown sample number
Mass sample +container
Volume of water (initial)
Volume of water (final)
Mass of empty container
.118 nm
Precision of Balances
Mass of Object
Other student’s a
measurements
Calculations Show your work using significant figures
1. a. Calculate the area of the foil .
b. Using the density and mass of the foil, calculate its volume.
c. Calculate the thickness of the metal foil in cm and them convert to nanometers
d. Calculate the diameter of an atom of aluminum in nanometers.
e. Using your answer from d. calculate the number of aluminum atoms that are stacked up and equal the thickness.
2.a. Calculate the volume of the substance in Procedure 2.
b. Determine the density of the substance from its mass and volume. c. Identify your substance, using Table 1 to compare
the densities. d. Calculate the percent accuracy error in your density measurement.
3. a. Use the class data to determine the average mass of the class object. b. Find the average precision error in the
measurement of class object.