Measuring Boltzmann`s constant with carbon dioxide

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Measuring Boltzmann’s constant with carbon dioxide
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2013 Phys. Educ. 48 713
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PAPERS
iopscience.org/ped
Measuring Boltzmann’s constant
with carbon dioxide
Dragia Ivanov and Stefan Nikolov
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
Abstract
In this paper we present two experiments to measure Boltzmann’s
constant—one of the fundamental constants of modern-day physics, which
lies at the base of statistical mechanics and thermodynamics. The
experiments use very basic theory, simple equipment and cheap and safe
materials yet provide very precise results. They are very easy and quick to
perform, allowing this fundamental constant to be brought to the
high-school level as a hands-on experience.
Introduction
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.
Theoretical considerations
We start off with the basic ideal gas law
(1)
pV = NkT,
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
pV
.
(2)
NT
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,
k=
m
.
(3)
M
Putting (3) into (2), we finally obtain the
working formula,
N=
pVM
.
(4)
mT
A new difficulty arises, however, with
measuring the mass of the gas. The experiment
can be carried out with different gases, but highly
c 2013 IOP Publishing Ltd
0031-9120/13/060713+05$33.00 k=
PHYSICS EDUCATION
48 (6)
713
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.
First experiment
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
NaHCO3 + CH3 COOH
→ CH3 COONa + H2 CO3 .
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
NaHCO3 + CH3 COOH
→ CH3 COONa + H2 O + CO2 .
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
MNaHCO3 ≈ 84 u = 139.44 × 10−27 kg
≈ 14 × 10−26 kg.
Similarly, the mass of a molecule of acetic
acid is
MCH3 COOH ≈ 60 u = 99.6 × 10−27 kg
≈ 10 × 10−26 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
1×10−3
m
21
= 14×10
NNaHCO3 = MNaHCO
−26 ≈ 7.1 × 10
3
molecules. Common vinegar is usually 6% acetic
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PHYSICS EDUCATION
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 cm3 of vinegar contains 0.6 g of
acid, which gives us approximately NCH3 COOH =
0.6×10−3
m
21
MCH3 COOH = 10×10−26 = 6×10 molecules. Thus,
for every gram of baking soda we should at the
very least use (7.1/6) × 10 cm3 ≈ 12 cm3 of
vinegar to ensure the release of all the CO2 . It
is advisable to use more, just in case—at least
20 cm3 —we actually used 30 cm3 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
7.1 × 1021 × 1.38 × 10−23 × 300
NkT
=
V
1 × 10−3
4
≈ 3 × 10 Pa = 0.3 atm.
p=
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 cm3 (with the part taken by the
plug already subtracted). The connective tubing
was measured and calculated to have a volume of
about 5 cm3 , which was added to the total volume.
We used either 20 or 30 cm3 of vinegar in the
experiments we quote in table 1; thus, our final
volume was 1075 or 1065 cm3 .
Measurement of the pressure is an important
part of the experimental setup. Depending on
the available instruments, we need to calculate
November 2013
Measuring Boltzmann’s constant with carbon dioxide
Table 1. A typical set of results from three experiments, with relative and absolute precision.
No.
m × 103
(kg)
p
(mmHg)
p × 103
(Pa)
V × 103
(m3 )
t
(◦ C)
T
(K)
k × 1023
(J K−1 )
1k/k
(%)
1k×1023
(J K−1 )
1
2
3
1.007
0.508
0.826
206
104
168
27.46
13.86
22.39
1.065
1.075
1.065
23.4
23.0
21.5
296.55
296.15
294.65
1.366
1.381
1.366
1.2
2.3
1.4
0.016
0.031
0.020
t °C
Figure 1. A sketch of the general setup.
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 CO2 , 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
November 2013
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.
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.
PHYSICS EDUCATION
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D Ivanov and S Nikolov
Figure 3. Filling the flask with carbon dioxide.
Figure 4. Pumping some of the carbon dioxide out.
This, again, ensures that we capture all of the
CO2 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
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 CO2 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 CO2 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 CO2 -filled flask is weighed with a
precise set of scales (our resolution was 1 mg),
giving us m1 . 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 m2 .
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 = m1 − m2 . Thus, we have all the values
1p 1V
1T
1m 1MNaHCO3
1k
=
+
+
+
+
.
k
p
V
T
m
MNaHCO3
Here, the individual precisions are as follows:
1p = 2 mmHg, 1V = 0.001 l, 1T = 0.1 K,
1m = 0.001 g, 1MNaHCO3 = 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 .
Second experiment
We have laid out the general setup for this method
in more detail elsewhere [6]. Here, we again use
soda and vinegar to produce CO2 . We mix them
716
PHYSICS EDUCATION
November 2013
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
References
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,
MCO2 = 44.01 u ≈ 73.06 × 10−27 kg
= 7.306 × 10−26 kg.
A typical set of results is m1 − m2 = 0.576 g,
V = 322 cm3 , patm = 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 .
Conclusion
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
November 2013
[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.
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.
PHYSICS EDUCATION
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