P
A P E R S iopscience.org/ped
Faculty of Physics, Plovdiv University ‘P Hilendarski’, 24 Tsar Asen strasse, 4000 Plovdiv,
Bulgaria
E-mail: draiva@uni-plovdiv.bg
and stnikolov@uni-plovdiv.bg
There are a number of fundamental constants at the core of modern physics—the speed of light,
Plank’s constant, the elementary electrical charge,
Boltzmann’s constant, and a few others. They are important both for physics as a science and for physics as a school subject. Their experimental determination is very important for the purposes of education but it is usually difficult in both theoretical and practical aspects.
Boltzmann’s constant is the most important constant in statistical mechanics and thermodynamics. In this paper we propose two school-grade laboratory experiments to measure
Boltzmann’s constant with easily obtainable, cheap and safe equipment and materials. Previous articles on ‘classroom’ measurement of
Boltzmann’s constant do exist [ 1 – 5 ], but they
require complicated and expensive equipment and
use advanced theory [ 4 ] without providing good precision (in [ 5 ], for example, getting the order
of magnitude right is considered a great success).
In comparison, our experiments are theoretically simple, quick to perform and cost virtually nothing, while providing very good precision.
We start off with the basic ideal gas law pV = NkT
,
(1) where p is the pressure, V is the volume and T is the temperature of the gas, N is the number of gas molecules and k is Boltzmann’s constant.
From ( 1 ) we obtain
k = pV
NT
.
(2)
The problem in determining Boltzmann’s constant using this simple formula is in measuring the number of molecules. It is usually obtained indirectly by measuring the mass of the gas, m , and dividing by the mass of a single molecule, M ,
N = m
M
.
(3)
Putting ( 3 ) into ( 2 ), we finally obtain the
working formula, k = pVM
.
mT
(4)
A new difficulty arises, however, with measuring the mass of the gas. The experiment can be carried out with different gases, but highly
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D Ivanov and S Nikolov
volatile substances (diethyl ether, dry ice [ 2 ]) are
often used because they can be weighed more easily in their non-gaseous form. These methods have some well-known drawbacks.
Principal idea
To produce the necessary amount of gas inside the vessel we used the well-known chemical reaction of baking soda (sodium bicarbonate) reacting with acetic acid (common vinegar). The reaction is
NaHCO
3
+ CH
3
COOH
→ CH
3
COONa + H
2
CO
3
.
The sodium acetate produced stays in solution while the carbonic acid dissociates into water and carbon dioxide practically completely.
Thus we ultimately obtain the full chemical equation
NaHCO
3
+ CH
3
COOH
→ CH
3
COONa + H
2
O + CO
2
.
It must be noted that each molecule of sodium bicarbonate produces one molecule of carbon dioxide. Thus, we need to weigh the baking soda and calculate its number of molecules, which will be equal to the number of carbon dioxide molecules—just what we need.
The amount of vinegar is not crucial so long as it is enough to react with all of the soda.
From the periodic table (or internet) we find out the mass of a molecule of sodium bicarbonate in atomic mass units. Consequently, considering that 1 u = 1
.
66 × 10
− 27 kg, we obtain
M
NaHCO
3
≈ 84 u = 139
.
44 × 10
− 27 kg
≈ 14 × 10
− 26 kg
.
Similarly, the mass of a molecule of acetic acid is
M
CH
3
COOH
≈ 60 u = 99
.
6 × 10
− 27
≈ 10 × 10
− 26 kg kg
.
It could be useful to have some preliminary idea of molecule numbers so that we can better determine the necessary amounts of soda and vinegar. One gram of baking soda will contain
N
NaHCO
3
= m
M
NaHCO3
=
1 × 10
− 3
14 × 10
− 26
≈ 7
.
1 × 10
21 molecules. Common vinegar is usually 6% acetic acid by volume (some brands may be different, so checking in advance is advised). As the density of acetic acid is just over 1 g cm
− 3
, we can consider that 10 cm
3 of vinegar contains 0.6 g of acid, which gives us approximately N
CH
3 m
M
CH3COOH
=
0
.
6 × 10
− 3
10 × 10 − 26
= 6 × 10
21
COOH
= molecules. Thus, for every gram of baking soda we should at the very least use
(
7
.
1
/
6
)
× 10 cm
3
≈ 12 cm
3 of vinegar to ensure the release of all the CO
2
. It is advisable to use more, just in case—at least
20 cm
3
—we actually used 30 cm
3 for 1 g of soda.
It is also useful to have an idea of the pressure expected to be generated. For a 1 l vessel with 1 g of soda at a temperature of 27
◦
C (300 K for ease of calculation) we obtain
NkT p = =
V
≈ 3 × 10
4
7
.
1 × 10
21
× 1
.
38 × 10
− 23
× 300
1 × 10 − 3
Pa = 0
.
3 atm
.
All of these calculations need not be presented to students. We provide them for the benefit of teachers to help in planning the experiment.
Performing the experiment
In order to perform the experiment we need an appropriate vessel that can be closed tightly with a rubber plug. Any flask of approximately 1–2 l is fine. A tube running through the plug connects the vessel to the manometer. The volume of the flask is determined in advance by filling it with water and precisely measuring the volume of the water with a measuring cylinder. To improve the precision it is necessary to take into account some other factors. The connective tubing needs to be as short as possible and its volume must be determined and added to the volume of the flask.
The volumes of the part of the plug that enters the flask and the vinegar must be subtracted in order to obtain the working volume. Our flask had a volume of 1090 cm
3
(with the part taken by the plug already subtracted). The connective tubing was measured and calculated to have a volume of about 5 cm
3
, which was added to the total volume.
We used either 20 or 30 cm
3 of vinegar in the experiments we quote in table
1 ; thus, our final
volume was 1075 or 1065 cm
3
.
Measurement of the pressure is an important part of the experimental setup. Depending on the available instruments, we need to calculate
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Measuring Boltzmann’s constant with carbon dioxide
No.
1
2
3
Table 1.
A typical set of results from three experiments, with relative and absolute precision.
m ×
(kg)
1.007
0.508
0.826
10
3 p
(mmHg)
206
104
168 p ×
(Pa)
10
27.46
13.86
22.39
3
V
(m
×
3
)
1.065
1.075
1.065
10
3 t
(
◦
C)
T
(K) k × 10
(J K
− 1
23.4
296.55
1.366
23.0
296.15
1.381
21.5
294.65
1.366
)
23 1 k
/ k
(%)
1.2
2.3
1.4
1 k × 10
(J K
− 1
)
23
0.016
0.031
0.020
t ° C
Figure 1.
A sketch of the general setup.
Figure 2.
A picture of the actual experimental setup we used. The paper that contained the soda is visible at the bottom of the flask.
the necessary amounts of soda and vinegar, as shown above. We have performed the experiment with a U-shaped water manometer, but this can only measure small pressure differences and does not give good precision. We have had good success using the pressure gauge of a blood-pressure measuring apparatus. It is easily capable of measuring pressures of the order of magnitude previously calculated, which is about
200–220 mmHg. It measures pressure in mmHg, so the reading needs to be multiplied by 133.3
to convert to pascals. The results given in table
1
were measured thus.
It must be noted that we are only interested in the partial pressure caused by the CO
2
, while the vessel also contains air at atmospheric pressure.
Both the devices we have used are differential by design, so they in effect subtract the atmospheric pressure for us. If an absolute pressure gauge is used, atmospheric pressure has to be measured separately beforehand and subtracted.
The temperature inside the flask is easily measured with a digital thermometer stuck through the rubber plug (making sure that it is air-tight). Most thermometers measure in degrees celsius, so it is necessary to add 273.15 to get the absolute temperature. The setup is presented in figures
1
and
2 .
The vinegar is easily measured with a measuring glass and poured into the vessel first.
The soda needs to be weighed as precisely as possible on a piece of paper put on the scales.
It is then tightly wrapped in the paper to keep it from spilling and to make it easier to drop it into the flask. In this way it will also take some time to react with the vinegar and will give us time to plug the flask before the reaction and avoid the loss of any gas. It also allows for many such ‘doses’ to be prepared in one sitting at the scales and then kept for later use in a well closed vessel to protect from moisture. We wrote the mass of the soda on the outside of the paper after wrapping.
The soda package is put into the flask so that it is slightly jammed into the neck. As we put the plug in it pushes the packet down into the vinegar.
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D Ivanov and S Nikolov
Figure 3.
Filling the flask with carbon dioxide.
This, again, ensures that we capture all of the
CO
2 released into the vessel. Once plugged, the flask is taken lightly by the neck (so that it is not warmed up by our hands) and shaken carefully.
Once the reaction starts we keep shaking until the pressure stabilizes (indicating that the reaction is over).
One set of data obtained by a group of students with some help from us yielded the results presented in table
1 . The total measurement
precision was calculated from the precisions of the separate measurements by the formula
1 k k
=
1 p p
+
1
V
V
+
1
T
T
+
1 m m
+
1
M
NaHCO
3 .
M
NaHCO
3
Here, the individual precisions are as follows:
1 p = 2 mmHg,
1
V = 0
.
001 l,
1
T = 0
.
1 K,
1 m = 0
.
001 g,
1
M
NaHCO
3
= 0
.
001 × 10
− 26 kg. It is easy to see that the main source of imprecision is the manometer, giving us about 1–2% by itself.
This is still remarkably good given the simplicity
(both theoretical and practical) of the experiment.
As can be seen from table
1 , our results
are in very good agreement with the official value of Boltzmann’s constant, which is 1
.
38 ×
10
− 23
J K
− 1
.
We have laid out the general setup for this method
in more detail elsewhere [ 6 ]. Here, we again use
soda and vinegar to produce CO
2
. We mix them
Figure 4.
Pumping some of the carbon dioxide out.
up in much larger amounts (2–3 tablespoons of soda and more than 100 ml of vinegar) and let the dioxide go out through a hose into a second, open flask of about 250–300 ml (figure
3 ). There,
it gradually displaces the air (as it is heavier), and thus we get a full flask of CO
2 at local atmospheric pressure. When we are sure that all the air has been displaced (it is advised to wait for at least a few minutes, as long as CO
2 is being produced in the first flask), we close that flask tightly with a rubber stop that has a glass tube coming out, fitted with a stopcock. The stopcock is open at first when plugging the flask and is then closed; this ensures that the gas is at atmospheric pressure.
The CO
2
-filled flask is weighed with a precise set of scales (our resolution was 1 mg), giving us m
1
. It is then connected to a vacuum pump (we used a low-grade hand-cranked pump) and as much of the gas as possible is evacuated
(figure
4 ). The stopcock is then closed and the
flask is weighed one more time, giving us m
2
.
Afterwards, the flask is turned over and the end of the tube is dipped into a vessel full of water. The stopcock is opened and water rushes inside until the internal pressure equalizes with atmospheric pressure (figure
5 ). The flask is then opened and
the sucked-in water is carefully poured into a measuring cylinder. The volume of that water is equal to the volume of the evacuated gas at atmospheric pressure. The mass of the evacuated gas is m = m
1
− m
2
. Thus, we have all the values
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Measuring Boltzmann’s constant with carbon dioxide are being developed), but they are not intended for educational use.
Received 13 June 2013, in final form 18 July 2013 doi:10.1088/0031-9120/48/6/713
Figure 5.
Sucking up water into the partially vacuumed flask.
we need to use formula ( 4 )—the pressure is the
local atmospheric pressure (measured separately with a barometer), the temperature is room temperature (again measured separately), the volume is the volume of the water and the mass is the difference between the two readings of the scales.
The molecular weight is that of carbon dioxide,
M
CO
2
= 44
.
01 u ≈ 73
.
06 × 10
− 27
= 7
.
306 × 10
− 26 kg
.
kg
A typical set of results is m
1
V = 322 cm
3
, p atm
− m
2
= 0
.
576 g,
= 99
.
6 kPa, T = 296
.
2 K.
This gives us a value for Boltzmann’s constant of k =
(
1
.
37 ± 0
.
02
)
× 10
− 23
J K
− 1
.
[1] Horne M et al 1973 An experiment to measure
Boltzmann’s constant Am. J. Phys.
41 344–8
[2] Lee E 1975 Determination of Boltzmann’s constant Phys. Teach.
13 305
[3] Kruglak H 1989 Boltzmann’s constant: a laboratory experiment Am. J. Phys.
57 216
[4] Campbell H M et al 2012 Experimental determination of the Boltzmann constant: an undergraduate laboratory exercise for molecular physics or physical chemistry Am. J.
Phys.
80 1045
[5] Tuykodi B et al 2012 The Boltzmann constant from a snifter Eur. J. Phys.
33 455–65
[6] Ivanov D 1996 Measuring the speed of molecules in a gas Phys. Teach.
34 278–9
[7] Fellmuth B et al 2011 Determination of the
Boltzmann constant by dielectric-constant gas thermometry Metrologia 48 382
[8] Gaiser C and Fellmuth B 2012 Low-temperature determination of the Boltzmann constant by dielectric-constant gas thermometry Metrologia
49 L4
Dragia T Ivanov graduated in engineering physics from Sofia
University, Bulgaria and then gained a
PhD in physics didactics from St
Petersburg, Russia in 1977. He has been a reader in physics didactics at Plovdiv
University, Bulgaria since 1980, becoming a professor in 2006. His areas of interest include physics didactics and multimedia teaching aids. He has written a number of books on practical physics experiments.
Both experimental setups have turned out to be sufficiently accurate, given their simplicity.
The experiments are quick and easy to perform and only require laboratory equipment that is relatively commonly available plus some baking soda and vinegar. We just use one formula, which is included in many curricula, and only the most basic algebra, so it should be within the grasp of most students at high-school level.
There are indeed much more precise methods available for measuring Boltzmann’s constant
[ 7 , 8 ] for metrological purposes (and new ones
Stefan N Nikolov gained his BSc in engineering physics from Plovdiv
University in 2005 and his MSc in medical and nuclear physics in 2007. He started teaching physics as early as 2001
(aged 18) and has been doing so ever since (still only as a part-time activity).
He is currently employed at Plovdiv
University as a physicist (lab technician) and is working towards obtaining his
PhD in physics didactics. His areas of interest are teaching and communicating physics (and science in general) and he has given a number of one-off presentations on different physics topics on numerous occasions.
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