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Lotto

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- Homework Statement
- The balloon casing is made of an impermeable impervious substance with a surface density of ##\sigma##. If the casing is completely filled with helium, it is shaped like a sphere of radius ##r##. The empty casing is infused with a certain amount of helium.

Determine the helium mass interval for which the resulting force acts on the balloon upward (the pressure in the balloon may be greater than atmospheric).

The molar mass of helium is ##M_{He}##, the molar mass of air is ##M_a##, the atmospheric pressure is ##p_0## and the temperature is ##T##.

- Relevant Equations
- ##pV=\frac mM RT##

##V=\frac 43 \pi r^3##

I suppose that the temperature is the same for the helium as well as for the air. So

##\frac 43 \pi r^3 \rho g >m_{He}g+4\pi r^2 \sigma g##.

I would determine the density of air from ##p_0 \mathrm d V=\frac{\rho \mathrm d V}{M_a}RT##.

So

##m_{He}<4\pi r^2\left(\frac{p_0M_a r}{3RT}-\sigma \right)##.

For the minimum mass ##m_0## it stands

##\frac 43 \pi r^3 p_0=\frac{m_0}{M_{He}} RT##,

because the pressure inside has to be bigger or the same as the atmospheric pressure, otherwise the balloon's volume is zero. So finally

##\frac{4p_0 \pi r^3 M_{He}}{3RT} \leq m_{He}<4\pi r^2\left(\frac{p_0M_a r}{3RT}-\sigma \right)##.

Is it correct? I am not sure about the temperature, because the assignment doesn't say what temperature it is.

##\frac 43 \pi r^3 \rho g >m_{He}g+4\pi r^2 \sigma g##.

I would determine the density of air from ##p_0 \mathrm d V=\frac{\rho \mathrm d V}{M_a}RT##.

So

##m_{He}<4\pi r^2\left(\frac{p_0M_a r}{3RT}-\sigma \right)##.

For the minimum mass ##m_0## it stands

##\frac 43 \pi r^3 p_0=\frac{m_0}{M_{He}} RT##,

because the pressure inside has to be bigger or the same as the atmospheric pressure, otherwise the balloon's volume is zero. So finally

##\frac{4p_0 \pi r^3 M_{He}}{3RT} \leq m_{He}<4\pi r^2\left(\frac{p_0M_a r}{3RT}-\sigma \right)##.

Is it correct? I am not sure about the temperature, because the assignment doesn't say what temperature it is.