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WEIGHT OF AIR IN MY BALLOON
Barbara Rovšek1, Nada Razpet1,2
1Faculty of Education, University of Ljubljana, Slovenia
2Faculty of Education Koper, University of Primorska, Slovenia
Corresponding author’s e-mail: barbara.rovsek@pef.uni-lj.si
1 Motivation
We are so accustomed to (and vitally dependent on) living in air atmosphere that we
usually do not pay any attention to it. We say the bottle is empty although it is full of air.
We occasionally notice the presence of air, when the wind is blowing or when we run out
of oxygen in a closed room full of people. But there are situations, when a teacher in
school would like to direct the pupils’ attention specifically to air. When talking about the
air pressure there is no bypass possible to realize the air has its own mass.
The question that arises naturally is what the mass of the surrounding air is and how
to expose it clearly. In available literature the teacher can often find suggestions how to
measure the weight of air, and many of them include a balance and a balloon [1, 2].
However, if the experiment is performed as suggested, from the results one can seldom
determine the right order of magnitude for the mass of the air in the balloon. There are
some important circumstances that are usually completely ignored, although their
influence is large. We shall discuss the weighing of the air in detail and present results of
measurements as can be obtained in every modestly equipped school laboratory.
2 Weighting the air
The most straightforward method of weighting the air comes to one’s mind immediately
– one can put a balloon on a precise electronic balance and measure its weight. The
balloon should be weighted by electronic balance twice – in the first case the balloon is
empty, and in the second it is full of air, as shown in Figs. 1 and 2.
Fig 1. Weighting an empty balloon.
Fig 2. Weighting a balloon, full of air.
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The mass of air in the blown balloon seems obviously equal to a difference of
measured masses of the full and empty balloon,
air
m
emptyfullair mmm
−
=
. (1)
For the masses of the empty balloon and the balloon full of air we have obtained mempty =
(12.1 ± 0.05) g and mfull = (12.4 ± 0.05) g and from Eq. (1) it follows that the mass of the
air in the blown balloon is mair = (0.3 ± 0.1) g. We have shown that the air has a mass
indeed. But we want to know more than that. Is the obtained value reasonable and does it
have any precise meaning? We can connect that value with some other measurable
physical quantity, like pressure.
To understand the calculus presented in the following a teacher at least should be
familiar with the ideal gas equation. All science teachers in primary schools had passed
some chemistry and physics courses, so that should not be a problem for them. Pupils
however do not need to know all details of the calculus to understand the idea behind. We
shall address this point again at the end.
The question that follows is, what is the pressure p inside the balloon, if the mass of
the air inside is taken to be the measured mass mair = 0.3 ± 0.1 g. From the ideal gas
equation we obtain
TR
VMm
p
balloon
air
=. (2)
To calculate the pressure from Eq. (2) we need to know the average (kilo)molar mass of
air
M
(= 29 kg/kmol), the universal gas constant R (= 8314 J/(kmol K)), the temperature
of air inside the balloon (the same as the temperature of air in the room, 300 K), and the
volume of the blown balloon . We have measured the average circumference
balloon
V
rccc
π
22/)( 21 =+= of the blown balloon, as shown in Figs. 3 and 4, and then
calculate the approximate volume of the balloon as
23
balloon 102.1
3
4−
⋅== rV
π
m3 .
Fig 3. Measuring the smaller
circumference of the blown balloon.
1
c
Fig 4.
e b
Measuring the larger circumference
2
cof th lown balloon.
2
Now we can calculate the pressure p of air inside the blown balloon from Eq. (2), and it is
found to be (21 ± 7) mbar. Is it possible that the pressure of air inside the blown balloon
is smaller than the normal air pressure 1 bar around the balloon? Of course not, so what’s
wrong?
3 Measuring the pressure of air inside the blown balloon
To find the meaning of the calculated pressure we have also measured the pressure of air
inside the balloon, as shown in Figs. 5 and 6. The balloon was connected to U–tube, filled
with coloured water. The U–tube was opened at one end and tightly attached to the
balloon at the other end. The pressure at opened end is the normal air pressure, that is
around 1 bar, and the pressure at the other end is somewhat larger. The difference of
pressure at both ends of the U–tube is balanced by the hydrostatic pressure of higher
water column in the opened side of the U–tube. The water level in the opened side is
found to be Δh = 13.5 cm above the water level in the side attached to the balloon. The
higher water column contributes to the pressure mbar 13.5 Pa 1350
water =
=
Δ
=
Δ
hgp
ρ
,
so the pressure of air inside the balloon is mbar 13.5 bar1
0
+
=
Δ
+
=
pppair . How does
this value compare to the pressure, calculated from the measured mass of air and ideal
gas equation, which was (21 ± 7) mbar?
Fig 5. Measuring the pressure of air
inside the balloon with the U–tube,
filled with coloured water.
Fig 6. Hydrostatic pressure of the
higher water column balances the
higher pressure inside the balloon.
To find the meaning and connection between the both values of pressure, calculated
and measured, we have to consider an important phenomenon that should not be
neglected or forgotten at all, when someone is trying to measure the weight of air – the
buoyancy.
4 The buoyancy in air sometimes matters
Everyone would agree that the balance will show less if we stand on it when it is put at
the bottom of the pool, filled with water. Our weight in the water is the same as outside,
but we feel less heavy because of the buoyant force acting on us. The criterion that tells
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us, how much the buoyancy influences the specific phenomenon, is the ratio of densities
– density of water compared to the density of the body, that sinks or floats in the water. If
they are comparable, one must not forget to include the buoyant force in his or her
equilibrium equations.
Just the similar criterion for buoyancy in the air tells us, that the buoyant force in air
may not be negligible if the densities of air and the body in the air are comparable, as
illustrated in Fig. 7. We can write equilibrium force equation,
buoyantggg FFFFF
+
=
+
=balance
balloon)
in(air (balloon) (3)
Fig 7. Equilibrium conditions: the force of gravity is balanced by
the sum of the buoyant force (large) and the force of balance (small).
The buoyant force equals the weight of the air, displaced by the balloon. Taking the
density of air at normal conditions (ρ = 1,2 kg/m3) we find that mass of the air, displaced
by the balloon equals 14 g! Can be a mass of displaced air more than an order of
magnitude larger than the mass of compressed air in the balloon? What do we actually
measure when we weight the blown and the empty balloon?
The electronic balance shows us its force, acting on the balloon, Fbalance. When the
balloon is blown, this force is F1, see Eq. (4), and when it is empty, the force of balance is
F2, see Eq. (5). With the empty balloon the buoyant force can safely be neglected.
buoyantgg FFFF
−
+
=
balloon)
in(air
balloon)
(empty(blown)1 (4)
balloon)
(empty(empty)2 g
FF
=
(5)
When we subtract F2 (5) from F1 (4), we obtain
gmmmgFFFF buoyantg
Δ
=
−
=
−
=− )(
air)
(displaced
balloon)
in(air
balloon)
in(air (empty)2(blown)1 , (6)
and we see from Eq. (6) that what we have measured was Δm – it equals to mair from
Eq. (1). Since Δm was found to be only (0.3 ± 0.1) g, we see that there is only slightly
more air inside the balloon than in the same volume of space in the space around. The
pressure p, calculated from the ideal gas equation (Eq. (2), 21 ± 7 mbar) corresponds
rather to measured pressure difference Δp (= 13.5 mbar) than to the full air pressure
inside the balloon. The two values are not much different any more, considering rather
rude measuring procedures.
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5 Conclusions
The buoyancy in surrounding air should not be neglected, when one is trying to weight
the air. It is possible to get reasonable values of mass of air inside the balloon only when
the buoyant force is taken into consideration, because it is almost as large as the gravity
force on the air inside the balloon. There is only a little bit more air in the blown balloon
than in the same volume of space in surroundings.
In general the buoyancy is not negligible, when the density of the object is of the
same order of magnitude as the density of the medium around. This is very common and
obvious statement, and nobody would forget to consider buoyant force on objects floating
in water. However, if the medium is air, the buoyant force usually is not considered at all,
since it has negligible influence on measurements of common weights. We are so
accustomed to neglect the buoyancy in air that we neglect or forget it even when we
should not.
Pupils in primary school are not able to follow the presented algebraic analysis,
which includes the ideal gas equation. Nevertheless a science or a physics teacher should
clearly expose the relevance of buoyancy in described experiment, if one wants to get
some reasonable estimation for the mass of air in a balloon. A teacher may also point to
other interesting (almost everyday) observations, connected to the buoyancy in air –
hovering of the hot-air or helium balloons, for example. In Slovenia the concept of
buoyancy in water itself is presented at physics course at 8th grade of primary school
(pupils of ages 13 – 14).
If one wants to measure the mass of air more precisely and directly, one can always
perform a classical experiment with a rigid gas container, which can be filled with
additional air or depleted. The volume of the rigid container does not change and that
makes the buoyancy in air non-relevant. However our intention was to discuss an
experiment, which seems to be a simpler – and therefore more attractive – version of the
former. We indeed find it as such, at least considering the equipment needed.
In literature [3-5] and on the web [6-8] there can be found many discussions on the
same or similar subject and we have listed a few in References. We found especially
interesting a suggestion in [7], where buoyant force is eliminated by reduction of the
balloon’s volume when pouring the liquid nitrogen over it. One needs to have the liquid
nitrogen though…
6 References
[1] B. Walpole and J. Ferbar, Veselje z znanostjo, Zrak, Pomurska založba (translation of
Air, Grisewood & Dempsey Ltd. 1987).
[2] N. Ardley, Spoznavajmo znanost, Zrak, Slovenska knjiga 1997 (translation of The
Science Book of Air, Harcourt Children’s Books 1991).
[3] M. J. Clouter, Archimes’ Principle: A Classroom Demonstration with a Twist, Phys.
Teach. 44, 46 (2006)
[4] T. B. Greenslade, Jr, The buoyancy balance: Nineteenth Century textbook
illustrations—LIV, Phys. Teach. 31, 160 (1993)
[5] U. Besson, Do things weigh more or less in the mountains? Phys. Educ. 41, 391
(2006)
[6] http://www.srh.noaa.gov/jetstream/atmos/ll_airweight.htm (7. 3. 2008)
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[7] http://www.millersville.edu/~physics/exp.of.the.month/64/ (7. 3. 2008)
[8] http://www.physics.isu.edu/~shropshi/smact.htm (7. 3. 2008)
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