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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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