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## Homework Statement

A helium balloon is used to lift a load of101N. The weight of the envelope of the balloon is46.5N and the volume of the helium when the balloon is fully inflated is31.5m3. The temperature of the air is 0°C and the atmospheric pressure is 1.00 atm. The balloon is inflated with a sufficient amount of helium gas that the net upward force on the balloon and its load is32.1N. Neglect any effects due to the changes of temperature as the altitude changes.

a) How many moles of helium gas are contained in the balloon?

b) At what altitude will the balloon be fully inflated?

(I'm still stuck on part a)

## Homework Equations

PV = nRT; n = PV/RT

V = m/d

## The Attempt at a Solution

If we solve PV=nRT for the number of moles, that should give us the number of moles of helium.

We are given the pressure of 1 atm, temperature of 0 C, and R is constant. We are not given the volume.

Converted to consistent units, we have:

P = 101325 Pa

T = 273.15 K

R = 8.314 J/mol*k

V = ? m^3

So, we need to find the volume.

V = m/d

We can look up the density of helium, but the mass is not directly given.

d = 0.179(kg/m^3)

We can calculate the mass by using the net force. The Archimedes Principle says that the net upward force (buoyant force)is equal to the weight of fluid displaced. The fluid displaced is equal to the fluid of helium inside the balloon.

The net upward force is 32.1 N

F = ma

m = F/a

m = 32.1N / 9.8(m/s^2) = 3.276 kg

V = m/d = 3.276kg / 0.179(kg/m^3) = 18.299 m^3

V = 18.299 m^3

So, now we have P, T, R, and V

n = PV/RT

n = (101325 Pa)( 18.299 m^3) / [ (8.314 J/mol*k)(273.15 k) ]

n = 816.45 mol

So, I get 816 mol, and the answer given for the problem is 734 mol.

I feel like I've checked pretty carefully to make sure units are in agreement and that I'm using the right values for constants. I also feel like the approach is correct, especially since the hint after submitting my answer said to use the Archimedes Principle to find the volume. I've been working on this problem for an embarrassingly long time, so I'd appreciate fresh eyes on it.

For reference, this problem is in the thermodynamics portion of physics 3.