MEASURING ATOMIC MASS
Purpose
The purpose of this experiment is to determine the mass of a copper atom
Theory
Nanotechnology involves working with objects in the range of one-billionth of a meter, or
about 100,000 times smaller than the thickness of a hair. One of the challenges of
working at very small sizes is making accurate measurements. Objects that we encounter
in our day to day lives can be easily weighed with a scale. But how can we measure the
mass of an atom? In this experiment we will apply basic principles of physics and
chemistry to indirectly determine the mass of a copper atom.
When trying to solve a difficult problem, it is sometimes useful to think about it in a
different way. Rather than trying to weigh an individual atom, suppose we could weigh a
large number of atoms and divide the result by the number of atoms. Determining the
mass of a very large group of atoms is easy, since we can just weigh them using a scale.
But now we have changed the problem to determining the number of atoms.
Electroplating provides us with a way to “count” the atoms.
Electroplating is an electrochemical process utilizing a chemical cell . The cell consists of
electrodes with a voltage applied across them. Between the electrodes is a solution that
contains positive and negative ions that move from one electrode to the other due to the
applied voltage. Ions are atoms or molecules that have gained or lost one or more
electrons. For example, a molecule that gains an extra electron has a charge of negative
one. An atom that loses 2 elections, like the copper atoms in this experiment, will have a
charge of +2.
In this experiment we will use a solution of copper sulfate (CuS0 4 ) in a chemical cell.
The solution contains positive copper ions (Cu +2 ) and negative sulfate ions (SO 4 -2 ). Metal
copper electrodes (which have no charge of their own) will be placed in the solution and
connected to an electrical power supply, The electrode connected to the positive side of
the power supply, known as the anode, will become positively charged. It will thus attract
the negative ions and repel the positive ions. The electrode connected to the negative side
of the power supply, known as the cathode, will become negatively charged, and will
attract positive ions and repel negative ions.
In the schematic diagram below, “A” represents a meter for measuring current (known as
an ammeter), “B” represents a battery, or power supply, and “C” represents the chemical
cell. R 1 , R 2 , and R 3 represent resistors, and are used to adjust the amount of current going
through the circuit. By creating a complete circuit and connecting it to a power supply, we
cause a current to flow in the circuit. Measuring current provides the key to achieving the
purpose of this experiment.
1
Anode (+)
Cathode (-)
red
A
black
C
R3
1
+
R1
10 
2
R2
6 V dc
B
Current is defined as the flow rate of charge, giving:
I =Q/t
(1)
where I is current in Amperes, t is time in seconds, and Q is charge in Coulombs. A
Coulomb can be thought of as a very large number of electrons. Current (I) is easily
measured with an ammeter, and time (t) is easily measured with a stopwatch. If we keep
the current constant for a specific amount of time, we can solve for charge as:
Q = It
(2)
Equation (2) gives the amount of charge, in Coulombs, transferred around the circuit. The
charge transferred through the electroplating cell in a given period of time must be equal
to that transferred through the ammeter and to that transferred through any other part of
the circuit.
But we’re not interested in coulombs. We want to know how many electrons passed
through the circuit. Just as we would find out how many eggs are in 4 dozen eggs by
multiplying the 4 dozen eggs by 12 eggs per dozen, we can find out the nu mber of
electrons by multiplying the number of coulombs by the number of electrons per coulomb.
There are 6.24 × 10 18 electrons per coulomb. We’ll use the symbol N e to represent the
number of electrons that move through the circuit.
N e  Q  6.24  1018
(3)
Each electron has a charge of -1, while each copper ion has a charge of +2. Each time a
copper ion is deposited on the cathode, it combines with two electrons. Where do the
electrons come from? At the anode, another chemical reaction takes place that releases 2
electrons, which flow through the wires, are measured by the meter, and then go to the
cathode where they combine with a single copper ion. Thus, the number of copper atoms
deposited on the cathode (which we’ll call N a ) is equal to half the number of electrons that
flow through the circuit, as shown in equation (4).
Na 
1
Ne
2
(4)
2
Now that we know the total number of copper atoms, we can calculate the mass of a single
atom by dividing the increased mass of the cathode by the number of atoms. The mass of a
single copper atom can now be calculated using equation (5). M meas in the equation below
is the difference between the mass of the cathode before and after electroplating.
M Cu =
M meas
M
 meas
number `of `atoms`transferred
Na
(5)
In order to check on the accuracy of our experiment, it is useful to have an alternative
answer for comparison purposes. The mass m of one atom can be calculated from
equation (6), given the molar mass A and Avogadro's number N o. One mole of an element
has a mass of A grams by the definition of the mole, and it contains N o (6.02 × 10 23 )
atoms.
m

A
No
(6)
We can then compare our experimentally determined mass with the generally accepted
value for mass, as found using equation (6). Use equation (7) to calculate the percent
error.
M meas  M pred
% error =
(7)
 100%
M pred
A note about working with very small and very large numbers
In this experiment we will be working with some very small numbers, such as the mass of
a copper atom. We will also be working with some very large numbers, such as the
number of copper atoms involved.
In order to express these numbers in a convenient, and easy to use format, scientists use
scientific notation. In scientific notation, numbers are divided into two parts. The first
part represents the specific numbers, and the second part represents the number of zeros
that show what is known as the magnitude of a number. The first part is always expressed
as a number as small as one, but smaller than 10. The second number is always expressed
as 10 to some power.
Each time a number is multiplied by 10, the decimal point is moved one place to the right.
For example, a number like 34000 would be written as 3.4 × 10 4 . The number 3.4 is
multiplied by 10 four times, thus the decimal point is moved over four places.
Very small numbers can be represented using negative exponents. 10 -1 means to divide by
10 (instead of multiplying by 10). So the number 3.4 × 10 -1 means we divide by 10, which
gives us 0.34, with the decimal point moved one place to the left. The number 3.4 × 10 -4
means we divide 3.4 by 10 four times, moving the decimal point 4 places to the left,
giving the number 0.00034.
3
Apparatus
Copper-plating cell
Copper sulfate solution
Variable Resistor, 10 ohm with 2 amp fuse
Variable Resistor, 2 ohm
Resistor, 1 ohm
Digital multimeter
Analytical balance
Isopropyl alcohol
Emery paper
Stopwatch
Procedure
1. Connect the apparatus according to the schematic drawing shown on page two of this
instruction. Use the center plate of the copper-plating cell as the cathode and the outer
pair of plates as the anode. Have the laboratory assistant check your wiring before
proceeding.
2. Fill the copper-plating cell about three-quarters full of copper sulfate solution. (This
may already have been done.)
3. Turn the knob on the 10 ohm rheostat fully counter-clockwise. Set the knob on the 2
ohm rheostat to its center position. Plug the power supply into a 120 Vac outlet, turn
its power switch on, and adjust the knob on the power supply for a voltage of roughly 6
volts. Adjust the 10 ohm variable resistor (and the 2 ohm variable resistor, if
necessary) for a current of approximately 1 ampere. (Note that if the meter is set for
milliamps, a reading of one amp will correspond with a reading of 1000 milliamps.)
4. Turn off the power supply and remove the center plate from the copper plating cell.
Clean the plate with fine emery paper and rinse it with water and alcohol. Be sure your
hands are clean and dry when handling the plate, and hold it only by the edges.
5. Air dry the plate. When the plate is dry, weigh it to the nearest milligram on the
analytical balance. Follow the instructions of the laboratory instructor carefully in
using the analytical balance. It can be easily damaged. Record the initial mass of the
plate on the data sheet.
6. Reinsert the plate into the cell without touching the portion of the plate to be
immersed. Clamp it firmly in place.
7. Turn on the power supply and start the stop watch at the same time. Use the rheostats
to keep the current constant for approximately 30 minutes. Note the exact time on the
stop watch when the power is turned off. Record the value of the current in amperes
and the time interval in seconds on the data sheet.
4
8. Lift off the top of the cell and remove the center plate carefully. Rinse it first in clean
water, then in alcohol, and air dry without rubbing. Handle it carefully so that none of
the copper that has been deposited on it will be rubbed off.
9. Weigh the plate to the nearest milligram, using the same analytical balance as for
the initial weighing. Record the final mass on the data sheet. Subtract the initial
mass to find the measured mass of copper deposited on the center plate (M meas ).
10. Complete the data table.
5
Measuring the Mass of an Atom: Data Sheet
Experimental Measurements:
Origninal mass of center plate
grams
Final mass of center plate
grams
Electric Current (I)
amps
Time (t) (make sure to convert to seconds)
seconds
Calculated Results
Mass of copper deposited
(M meas )
grams
Charge transferred
Coulombs
Number of electron charges transferred (Ne)
.
Number of atoms transferred (Na)
.
Mass of one copper atom (experimental)
grams/atom
Mass of one copper atom (eq. 6)
grams/atom
Percent error
%
6