How many circus balloons will it take to lift a 80.0 kg person?

The pressure is 1.00 atmospheres. The temperature is 298 K. The air is composed of 20.0% oxygen and 80.0% nitrogen.

Each balloon is a perfect sphere, 40 cm in diameter, filled with helium gas (also at 1.00 atmospheres). The mass of the balloon rubber and string for each balloon is 1.00g

In order to find how many balloons we need to lift an 80 kg person we start by finding how much weight one balloon can lift. To do this we find the net force acting on a single balloon. The balloon is only acted on by the buoyant force (upwards), and the force of gravity (downwards), thus:

F
n
e
t
=
F
B

F
g
=
ρ
a
i
r
g
V

(
m
+
ρ
H
e
V
)
g

Here we have used the fact that the mass of the balloon is m=1 gram plus the mass of the helium gas, which is the density of helium multiplied by the volume. Rearranging this gives:

F
n
e
t
=
(
ρ
a
i
r

ρ
H
e
)
g
V

m
g

We can find the density of air as
ρ
a
i
r
=
0.2
ρ
o
x
y
g
e
n
+
0.8
ρ
n
i
t
r
o
g
e
n
where the densities of oxygen and nitrogen are of the diatomic molecules at the temperature and pressures we want. Since only our temperature is different from standard temperature and pressure (STP) we can find the new densities as:

ρ
=
ρ
s
t
p
×
273
T

With T being the temperature we wish to use. This gives a density of diatomic oxygen of 1.31 kg/m
3
and a density of diatomic nitrogen of 1.15 kg/m
3
. Thus our density of air is 1.18 kg/m
3
. We can also find the volume of one spherical balloon as:

V
=
4
π
3
r
3
=
4
π
3
(
0.2
)
3
=
0.0335
m
3

Now we may calculate our force due to one balloon (density of He is 0.163 kg/m
3
).

F
n
e
t
=
0.324
N

Now that we know the lifting force of one balloon, we can take the total force we need to lift (the weight of a 80 kg person), and divide by this lift to find how many balloons we need.

F
g
=
M
g
=
(
80
kg
)
(
9.8
)
=
784
N

The number of balloons we need (x) is then:

x
=
F
g
F
n
e
t
=
784
0.324
=
2420

Thus we would need 2420 of these circus balloons to lift a 80 kg person

To determine how many circus balloons it will take to lift a 80.0 kg person, we need to consider the buoyant force generated by the balloons and compare it to the weight of the person.

The buoyant force is given by the difference between the weight of the air displaced by the balloon and the weight of the balloon itself. In this case, we'll assume that the entire volume of the balloon is filled with helium gas, which is lighter than air.

First, let's calculate the weight of the person:
Weight = mass × acceleration due to gravity
Weight = 80.0 kg × 9.8 m/s²
Weight = 784 N

Next, let's calculate the weight of an individual balloon (without considering the rubber and string):
Weight of balloon = density of air × volume of air displaced × acceleration due to gravity

The density of air can be calculated using the ideal gas law:
PV = nRT
n = (P × V) / (R × T)
n = (1.00 atm × V) / (0.0821 L×atm/(mol×K) × 298 K)
n = (V / 24.708) mol

Since the air is composed of 20.0% oxygen and 80.0% nitrogen, we can calculate the weight of this mixture using the molar masses of oxygen (O₂ = 32.00 g/mol) and nitrogen (N₂ = 28.02 g/mol):
Weight of air mixture = (0.20 × 32.00 g/mol + 0.80 × 28.02 g/mol) × n
Weight of air mixture = (6.40 g/mol + 22.42 g/mol) × (V / 24.708) mol
Weight of air mixture = (28.82 g/mol) × (V / 24.708) mol
Weight of air mixture = 1.168 g × V

To find the volume of a balloon, we need to calculate its radius:
Radius = diameter / 2
Radius = 0.40 m / 2
Radius = 0.20 m

Volume of balloon = (4/3) × π × radius^3
Volume of balloon = (4/3) × 3.14159 × (0.20 m)^3
Volume of balloon = 0.03351 m³

Now, let's calculate the weight of an individual balloon:
Weight of balloon = (1.168 g × V) × 0.001 kg/g × 9.8 m/s²
Weight of balloon = 0.01145 V kg

Since each balloon weighs 0.01145 V kg, the buoyant force it generates is equal to that weight. So, to counteract the weight of the person (784 N), we need:
Number of balloons = Weight of person / Weight of each balloon
Number of balloons = 784 N / (0.01145 V kg)

Now, we can substitute the given diameter of the balloon to calculate the number of balloons needed to lift the person:
Number of balloons = 784 N / (0.01145 × (4/3) × 3.14159 × (0.20 m)^3 kg)

Using this equation, you can substitute the given values and calculate the answer.