Number of balloons filled from a tank of helium

[Mitch_11]

Homework Statement

How many 3L balloons can you fill from a 20L tank of helium at a pressure of 14.4atm.

Homework Equations

Boyle's Law: [tex]P_{1}V_{1}=P_{2}V_{2}[/tex]

The Attempt at a Solution

[tex]\frac{14.4atm\times20L}{1atm}=V_{2}[/tex]

I then divided the answer to thank by 3 to get the amount of balloons. However my teacher has said this is incorrect. He said there is a small 'nuance' in the problem that must be considered. The only thing I can think of is to increase the initial pressure by 1 as the gauge may not have added atmospheric pressure to the reading. I am just wondering if this is correct or if there is something I have overlooked.

Thanks for any assistance.

 

[Dickfore]

If you fill one balloon, the pressure in the tank drops.

 

[Dickfore]

The answer i get is eighty nine.

 

[Mitch_11]

What equation did you use to work that out?

 

[Dickfore]

First of all, Boyle's Law only holds for a given fixed quantity of gas. Here, this is not so. You should use the full ideal gas Law (Clapeyron's formula):

[tex]
P V = N k T
[/tex]

where N is the number of molecules in the gas, T is the absolute temperature and k is the Boltzmann constant.

From here, I worked out how many molecules of helium are there in helium baloon at a given temperature (in symbolic form). Obviously, for x such baloons, you would need x times more molecules. Although you do not know T, it cancels in the final result.

Then, I worked out how the pressure in the tank would change if we removed that number of molecules at the same temperature. The pressure can drop from the original value only to the pressure in the balloon. You cannot inflate a balloon with a tank at pressure lower than the required one. The result should follow from here.

 

[Mitch_11]

Ok I understand now. Thank you for explaining that.

 

[Borek]

[tex]
P V = N k T
[/tex]

That's not incorrect, but I prefer

[tex]P V = n R T[/tex]

for practical reasons - you can easily find R in any units you need, while k is usually expressed in J/K, and number of moles is usually much easier to work with then the number of atoms (even if these are easily interconverted).

 

[Mitch_11]

Ok. Here is what I did.

First, as you said, I worked out the number of moles of helium needed to fill each balloon to 3L. Assuming that 1 mole is 24.47L at 25^oC (298K) I calculated that each balloon needs 0.12moles of gas to fill it.

I then used [tex]PV=nRT[/tex] to work out that there are 11.8 moles of gas in the cylinder.

From here I worked worked out the amount of moles needed to keep the cylinder above 1atm, therefore still allowing helium to go into the balloon. I worked this out to be 0.8 moles.

I then subtracted 0.8 from 11.8 to get the number of moles that will go into the balloons. I then divided this by 0.12 to work out how many balloons could be filled.

My answer worked out to be 91 (rounded). If I have gone about this the wrong way and just got the answer from luck, could you please show where I have gone wrong.

Thanks

 

[Borek]

You are not given temperature and in fact it is irrelevant here. All we have to do is to assume that both balloons and tank have exactly the same pressure.

You were right about using Boyle's law, and you were right you have to calculate volume of the gas after decompression. However, you have failed to see that gas after decompression has to fill not only balloons - 20L have to stay in the tank!

And IMHO 89 is not a correct answer, that's another nuance

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