How Much Hydrogen Is Needed for a Balloon to Float at 10km Altitude?

[cycrups]

Homework Statement

1. At an altitude of about 10 km, the temperature of the Earth’s atmosphere is roughly -50°C and the air pressure is around 16 kPa. (Assume ideal gas)
   
   1. How many kilograms of hydrogen gas (H2, molecular mass 2g/mol) should be put in a balloon to fill it to 2000 m3?
   2. What is the required mass of total ballast that the balloon will float at constant height (molecular mass of air 29 g/mol)?
      

Homework Equations

PV = nRT

The Attempt at a Solution

For 1. I used V = (P*A*N)/(R*T) = ( 2g/mol * 16000 Pa * 2000m^2 ) / (8.314 * 223K) = 34.51 Kg I don't know if it's correct way to do it or not

For the second part I have no idea how to even start.

 

[Orodruin]

For 1. I used V = (P*A*N)/(R*T) = ( 2g/mol * 16000 Pa * 2000m^2 ) / (8.314 * 223K) = 34.51 Kg

Your numbers make sense, although the notation in your equation could be clearer. V is generally used to denote a volume, not a mass.

For the second part I have no idea how to even start.

What must be fulfilled in order for the balloon not to accelerate either up or down?

 

[cycrups]

Your numbers make sense, although the notation in your equation could be clearer. V is generally used to denote a volume, not a mass.
What must be fulfilled in order for the balloon not to accelerate either up or down?

certain weight?

 

[Orodruin]

Naturally, but what determines this weight? What determines whether any object is accelerating or not? Hint: There was this guy in the 17th century who got hit by an apple ...

 

[cycrups]

Naturally, but what determines this weight? What determines whether any object is accelerating or not? Hint: There was this guy in the 17th century who got hit by an apple ...

Force. Hope it's correct.

 

[Orodruin]

Yes, so what must be true if the balloon is not supposed to accelerate?

 

[cycrups]

Yes, so what must be true if the balloon is not supposed to accelerate?

The forces in both directions must equal the same.

 

[haruspex]

I used V = (P*A*N)/(R*T)

I guess you mean M = (molecular mass)*n = (molecular mass)*PV/(RT)

The forces in both directions must equal the same.

Right. And without the ballast, what two forces do you have?

 

[cycrups]

I guess you mean M = (molecular mass)*n = (molecular mass)*PV/(RT)

Right. And without the ballast, what two forces do you have?

I'd say gravity and the force of the helium pushing up the balloon?

 

[haruspex]

I'd say gravity and the force of the helium pushing up the balloon?

How does the helium push the balloon up? Does it have negative mass??

 

[cycrups]

How does the helium push the balloon up? Does it have negative mass??

because Helium is lighter than air that's why it tends to push up

 

[Orodruin]

It would be more accurate to say that the surrounding air is pushing the helium up - with a force greater than the helium weight. If you block the helium from escaping, the surrounding air is providing the upward force on the balloon helium system.

To get back to the problem: What is the buoyancy force on the balloon + helium system? What forces act in the opposite direction? Does the weight of the system compensate for the buoyancy? If not, how much weight do you need to add?

 

[cycrups]

It would be more accurate to say that the surrounding air is pushing the helium up - with a force greater than the helium weight. If you block the helium from escaping, the surrounding air is providing the upward force on the balloon helium system.

To get back to the problem: What is the buoyancy force on the balloon + helium system? What forces act in the opposite direction? Does the weight of the system compensate for the buoyancy? If not, how much weight do you need to add?

Should I use this equation to get the Buoyant force F (b) = (air density)x(9.81 m/sec2)x(volume of the gas filled balloon) ?

 

[haruspex]

Should I use this equation to get the Buoyant force F (b) = (air density)x(9.81 m/sec2)x(volume of the gas filled balloon) ?

Yes.

 

[cycrups]

Yes.

or should I use Fb= mass of air * g = P * Vb * g because I don't have air density OR do I get the air density from

density = P / R * T ?

 

[haruspex]

or should I use Fb= mass of air * g = P * Vb * g because I don't have air density

mass = pressure * volume? Where does that come from?

OR do I get the air density from density = P / R * T ?

Sort of. You know air density at standard surface temperature and pressure, right? So use that to figure out what it would be at the given temperature and pressure.

 

[cycrups]

mass = pressure * volume? Where does that come from?
.

I took it from http://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/Chapter2b.html Problem 2.9

 

[cycrups]

mass = pressure * volume? Where does that come from?

Sort of. You know air density at standard surface temperature and pressure, right? So use that to figure out what it would be at the given temperature and pressure.

Okay so the density would be → density of air = 16000 Pa / 8.31 * 223 K = 8.63 Kg/m³

and then to find the Buoyant force Fb = 8.63 * 10 m/s³ * 2000 m³ = 172600

Is that correct, and if it is what's the next step? Do I equate Fb = Fg to get the Maximum altitude?

 

[haruspex]

Okay so the density would be → density of air = 16000 Pa / 8.31 * 223 K = 8.63 Kg/m³

and then to find the Buoyant force Fb = 8.63 * 10 m/s³ * 2000 m³ = 172600

Is that correct, and if it is what's the next step? Do I equate Fb = Fg to get the Maximum altitude?

Sorry, I wasn't paying enough attention to exactly what data you are given.
Here's the easiest way:
- you have already calculated the mass of H2 for the given volume, temperature and pressure
- you know the molecular mass of H2 and the molecular mass of air
- how can you simply combine those facts to obtain the mass of air at the given volume, temperature and pressure?

 

[cycrups]

Sorry, I wasn't paying enough attention to exactly what data you are given.
Here's the easiest way:
- you have already calculated the mass of H2 for the given volume, temperature and pressure
- you know the molecular mass of H2 and the molecular mass of air
- how can you simply combine those facts to obtain the mass of air at the given volume, temperature and pressure?

I equate both of them together so the PV/nRT (of H2) = PV/nRT of air? I am sorry I am really having a hard time solving this part.

 

[haruspex]

I equate both of them together so the PV/nRT (of H2) = PV/nRT of air? I am sorry I am really having a hard time solving this part.

PV/nRT is always 1, so that doesn't seem very profitable.
You have two equations of the form PV=nRT, one for air, one for H2: P_aV_a=n_aRT_a, P_hV_h=n_hRT_h. You know that some of the variables are the same in the two equations. Which ones? What does that tell you about the remaining variables?

 

[cycrups]

PV/nRT is always 1, so that doesn't seem very profitable.
You have two equations of the form PV=nRT, one for air, one for H2: P_aV_a=n_aRT_a, P_hV_h=n_hRT_h. You know that some of the variables are the same in the two equations. Which ones? What does that tell you about the remaining variables?

Temperature and R should be the same. The thing I don't know would be the air Pressure and Volume but I doubt the Air would have volume + it's not an ideal gas because it's air. Am I making sense?

 

[haruspex]

Temperature and R should be the same. The thing I don't know would be the air Pressure and Volume but I doubt the Air would have volume + it's not an ideal gas because it's air. Am I making sense?

We're talking about the mass of air displaced by the balloon. By definition, that will have the same volume as the balloon.
It is true that the pressure of the H2 will be slightly greater than that of the surrounding air, but only slightly or the balloon would rip. So if the pressures, volumes and temperatures all match, what do you deduce about n_a and n_h?