A weather ballon is prepared to lift off at the North Pole. The balloon is filled with 15000 L of helium gas. The conditions at the launch site are 760 mm Hg and 0 degrees celsius. The balloon is then released and rises to a height of 20,000 m. The atmospheric pressure decreases at a rate of 0.0032mm Hg per meter. The temperture drops at a rate of 0.001 celcius per meter. At 15,000 m the balloon reaches its maximum unpressurized volume of 82.5 kl, at which time the internal pressure begins to increase at a rate of 10mm Hg per 100L increase. The balloon ultimately reaches it maximum volume of 90kL at 19,500m. What is the density g/m of the helium gas at 20,000m? How many molecules of He2 are in the balloon (assuming no loss)?

To find the density of the helium gas at 20,000 m, we need to calculate the mass of the gas and divide it by the volume.

First, let's consider the changes in pressure and temperature with respect to altitude:

1. Pressure: The atmospheric pressure decreases at a rate of 0.0032 mm Hg per meter. So, at 20,000 m, the pressure will decrease by (0.0032 mm Hg/m * 20,000 m) = 64 mm Hg.

The new pressure at 20,000 m can be calculated by subtracting the decrease from the initial pressure:
New Pressure = 760 mm Hg - 64 mm Hg = 696 mm Hg.

2. Temperature: The temperature drops at a rate of 0.001 degree Celsius per meter. So, at 20,000 m, the temperature will decrease by (0.001°C/m * 20,000 m) = 20°C.

The new temperature at 20,000 m can be calculated by subtracting the decrease from the initial temperature:
New Temperature = 0°C - 20°C = -20°C.

Now, let's convert the volume and pressure to their respective SI units:

Volume: 82.5 kL = 82,500 L
Pressure: 696 mm Hg = 696/760 atm = 0.9168 atm

To calculate the mass of the gas, we can use the Ideal Gas Law equation:

PV = nRT

Where:
P = pressure in atm
V = volume in L
n = moles of gas
R = ideal gas constant (0.08206 L·atm/mol·K)
T = temperature in Kelvin

Since we want to find the density, we can rearrange the equation to solve for moles (n):

n = PV/(RT)

Substituting the known values:

n = (0.9168 atm * 82,500 L) / [(0.08206 L·atm/mol·K) * (-20+273.15) K]
n = 3261.304 mol

Next, we need to calculate the mass using the molar mass of helium:

Molar Mass of Helium (He) = 4.003 g/mol

Mass of Gas = Moles of Gas * Molar Mass of Helium
Mass of Gas = 3261.304 mol * 4.003 g/mol
Mass of Gas = 13,052.614 g

Finally, divide the mass by the volume to get the density:

Density = Mass / Volume
Density = 13,052.614 g / 82,500 L
Density ≈ 0.158 g/L or 158 g/m³

So, the density of the helium gas at 20,000 m is approximately 158 g/m³.

Now, to calculate the number of molecules of He2 in the balloon, we can use Avogadro's number (6.022 x 10^23 molecules/mol) and the number of moles we calculated earlier:

Number of Molecules = Moles of Gas * Avogadro's Number
Number of Molecules = 3261.304 mol * 6.022 x 10^23 molecules/mol
Number of Molecules ≈ 1.963 x 10^27 molecules

Therefore, assuming no loss, there are approximately 1.963 x 10^27 molecules of He2 in the balloon.