What helium mass in a balloon to make it bouyant in air?

[Lotto]

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.

 

[erobz]

Pardon my ignorance…Are they asking how much helium is required so the balloon floats?

 

[Lotto]

Pardon my ignorance…Are they asking how much helium is required so the balloon floats?

The question is what mass should the helium have so that the net force acting on it is oriented upward. So the buoyancy force can be greater that the gravity force of the balloon with the helium.

 

[erobz]

Did you add partial pressures of helium and air in the mixture?

 

[Lotto]

Did you add partial pressures of helium and air in the mixture?

No, because I suppose that there is only helium inside of the balloon.

 

[nasu]

Did you add partial pressures of helium and air in the mixture?

They say : "The empty casing is infused with a certain amount of helium."
So, it looks like there is only helium inside. Even though "empty" when it comes to gases may need clarification.

 

[Lotto]

So, is my solution correct or not?

 

[Hill]

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: $pV=\frac mM RT$
$V=\frac 43 \pi r^3$

... the assignment doesn't say what temperature it is.

It does not?

... the temperature is $T$.

 

[Lotto]

It does not?

I meant that it does not say if it is a temperature of the helium or the air.

 

[erobz]

I meant that it does not say if it is a temperature of the helium or the air.

I think you are to assume thermal equilibrium like you have.

 

[erobz]

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

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

This seems fine. Where $\rho$ is the density of the air

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

I'm not sure what you are doing with the $dV$'s in this part? The density of the surrounding air is just given by the Ideal Gas Law:

$$ p_o = \frac{ n_{air} RT }{V} = \frac{ \rho_{air} RT }{M_{air}} $$

$$ \rho_{air} = \cdots $$

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

This looks good.

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.

Looks reasonable to me.

 

[erobz]

Uh oh...I got a skeptical...what has been missed @hutchphd?

 

[hutchphd]

I rescinded it, sorry. Past my thinking hour.