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Why do helium-filled balloons float?

Why do helium-filled balloons float?

Our family does balloon launches on the 4th of July afternoon, instead of fireworks.
– Unknown

Why do balloons float?

Can you fill a balloon part way with helium, make an animal out of it and have it float?

Definitely in water. In air… well, that depends.

This is an application of “Archimedes’ Principle” which states that a body immersed in a fluid is buoyed up by a force equal to the weight of the fluid displaced. If an object immersed in a fluid is heavier than the fluid it displaces, it will sink to the bottom, and if lighter than the fluid it displaces, it will rise.

In our case, the “body” is a Helium-filled 260, and the “fluid” is air.

If we use the “Ideal Gas Law” we can quickly find the mass of the Helium or air in a 260. The “Ideal Gas Law says that m = PVM/RT where:

m = mass of the gas in grams
P = pressure inside the balloon, in atmospheres
V = volume in liters
M = Molecular weight of the gas
R = a constant = 0.082
T = Absolute temperature, Kelvin

At sea-level, the air pressure is 1 atmosphere and say we are at room temperature. Then P = 1 and T = 293K.

Then approximate a fully inflated 260 as a 1.75″ dia cylinder that is 50″ long. Since the pressure in the balloon is only slightly greater than the pressure outside the balloon, let’s call them the same for now (both equal to 1 atmosphere).

The volume is pi x 0.875″ x 0.875″ x 50″ long = 120.2 in^3 or:
1 liter V = 120.2 in^3 x ——— = 2.0 liters. 61 in^3
We next look up the molecular weight of the gasses:

M = 4.00 for Helium,
M = 28.97 for the mixture of gasses we call “air”

and plug all these numbers into the Ideal Gas Law to find:
1 x 2.0 x 4.00 the mass of Helium in a 260 = m = —————- = 0.33 grams 0.082 x 293
1 x 2.0 x 28.97 the mass of air displaced = m = —————– = 2.41 grams 0.082 x 293
OK, we’re almost done.

The mass of a 260 is 1.6884 grams (it’s nice to have a laboratory balance that weighs to a ten-thousandth of a gram… ), but let’s just call it 1.70 grams since we’re all friends here. Now add that to the 0.33 grams of Helium inside the balloon for a total weight:

The total mass of the “body” (balloon + Helium) is 2.03 grams

The mass of the fluid displaced (air) is 2.41 grams

An object’s mass doesn’t change, but it’s “weight” depends on the force of gravity. Converting these masses to equivalent weights on earth, the Helium-filled balloon weighs 0.0045 pounds and the displaced air pushes up with a weight of 0.0053 pounds. This results in a net buoyant force or “lift” of 0.0007 pounds on a Helium-filled 260, so it should float (but just barely!) !!!

So, let’s see if any of this works in the real world:

PFFFFffffffffffffffttt! PFFFFffffffffffffffttt! PFFFFffffffffffffffttt! PFFFFffffffffffffffttt! PFFFFffffffffffffffttt! PFFFFffffffffffffffttt! Squeak, squeek, SQUEEEEEAAAK, SQUeak, squEAK, squeak… Ta-dah!

Well, I now have a parasol floating above my head, so yes, fully inflated 260 sculptures with a minimal number of twists will indeed float (just not for very long…). This would be good for a quick demo, if nothing else. The Helium diffuses out rapidly, and there is just not enough reserve lift to keep it aloft as the 260’s start to shrink.

HOWEVER, Freda asked “Can you fill a balloon *part way* with helium and make an animal out of it and have it float?”

Yes, you could make an animal that will float for a short period of time if you inflate the balloon *most of the way*, leaving only a short nipple. Neglecting the lift (but not the weight) for the uninflated nipple, the math says that for neutral buoyancy (the “just” floating condition where the weight equals the lift):
0.33 g 2.41 g 1.7 g + ———— x (inflated length) = ———— x (inflated length) 50″ inflated 50″ inflated
Solving this equation for the “inflated length” term gives the minimum inflated length for buoyancy = 41″. So if you could fill a 260 up to 1.75″ diameter and make a figure that had a total bubble length of at least 41″, it should hang in the air for a while.

How do you get more than a few minutes of flying time out of a Helium-filled 260?

  1. inflate all the way and don’t twist it much.
  2. inflate to as large a diameter as you can (don’t burp it).
  3. cut off the rolled nozzle after you tie it (shed those milligrams!).

To get as large a diameter as possible for # 2, try this trick: put a paperclip across the nozzle of the uninflated 260 and drop it into some boiling water for a while. Take the balloon out of the water, remove the clip (which has kept any water from getting inside the balloon) and *immediately* inflate it with helium without even drying the outside. You’ll see that water has a huge effect on the properties of latex.

What is the effect of the latex compressing the helium?
You can fill a balloon, then you can fill it some more before it pops.
Assuming the balloon is not getting as much bigger as it had been
during inflation, the helium is more compressed.

I don’t know what the exact pressures are because I don’t have a pressure gage that measures fractions of a psi. I would have to use a manometer to measure the “head” (height that a water column is raised by the air pressure) in a 260 and then convert that to pressure.

Is it heavier for having more helium inside?

Yes. Consider how a tank full of compressed gas is heavier when full than when empty. Since we are talking about tenth-of-a-gram lift levels here, this can’t be neglected for a 260. However, if the extra pressure causes the diameter to grow a bit, then we will most likely see an increase in lift.

Where is the optimum lift on the inflation graph?

The lift is directly proportional of the volume of helium in the balloon.

The volume of helium is directly proportional to the length of the balloon. (A fully inflated 280 will have 33% more lift than a 260) The volume of helium is also directly proportional to the cross-sectional area of the 260.

BUT, the cross-sectional area of the 260 is proportional to the square of the balloon diameter.

So, the fastest way to increase the lift is to increase the diameter. That’s why I suggested soaking the balloon in boiling water before inflating. The water is absorbed into the latex and reduces its elastic modulus, allowing it to stretch more. It will stretch more in the direction of maximum stress, the circumferential or “hoop” direction, and you’ll see this as a diameter increase.

(No, I don’t know how much water is absorbed, and yes, the weight of the absorbed water does make the balloon heavier, and yes, there is probably an optimum amount of time to soak the balloon which maximizes the diameter increase to water-weight gain ratio… but that’s a little deeper than I wanted to get into this). Inflating with heated air may also result in a diameter increase.

Inflating & deflating a balloon several times will also increase it’s diameter. You wouldn’t have water weight gain to worry about either. It does tend to make the balloon thin-skinned & harder to work with, but the added volume of helium is well worth the effort.

To answer your original question, to find the optimum lift as a function of helium pressure, you need a graph of the balloon diameter against internal pressure. Given those numbers and the formulas above, you could find the pressure for optimum lift. However, it might all be an exercise in futility because:

  1. the wall thickness variation may exceed the pressure/diameter variation,
  2. the balloon weight variation may exceed the few tengths of a gram lift we have,
  3. more pressure makes the helium diffuse out faster (but overall I’d expect it to last longer – just not twice as long for twice the pressure)
  4. the optimum may be at the burst point, which would vary for each balloon depending on it’s surface and wall condition. Have you ever looked closely at an uninflated balloon? Each Qualatex 260 typically has a few particles embedded in the wall. In the paint industry, they call these specks of solid matter in a coat of paint or varnish “nibs”. In the balloons, I am not certain what the nibs are made of – they may be hardened latex, dirt, lumps of coloring agent, etc. and it would take a little work to find out exactly what they are. Upon inflation, a thin-walled region develops around some of the nibs. The size of the nib in the wall seems to determine whether or not there is a thin area around it (as I’m certain the specific nib material would as well, due to the interfacial bond with the latex of the balloon). The larger nibs cause greater stress concentrations which create the thin areas around them (from elementary solid mechanics principles). I would say that the balloon would burst at these thin areas first, and that the variation in the burst point may outweigh the pressure/diameter variation.

Are you saying that a twisted dog won’t float because the gas makes it too heavy to float?

No, just that the more gas you add, the heavier the balloon becomes. (It’s the same as adding more water to a water balloon. It’s true that air is a gas and water is a liquid; but they are both “fluids” in a technical sense and obey many of the same laws)

Also I was saying that I don’t know how much pressure a balloon will hold before it bursts, because I don’t have the equipment to measure pressures that small.

Part of Tom’s question was: once the balloon is fully inflated, does adding more Helium increase the lift? Well, if the balloon doesn’t get any bigger at all, the answer would be no. If the answer was that the balloon stretches a tiny bit bigger, then I’d have to know; did the tiny increase in size add enough lift to offset the extra weight of helium added. I don’t know the answer because latex properties are highly nonlinear, the stress state is complicated, and you really can’t calculate an answer – you just have to do an experiment and measure it.

When everyone says that the gas leaks out real fast, how fast is fast?
Like a normal balloon? Or in a few minutes?

I think that the gas leaks out at the same rate (per unit of surface area) as it would in a round balloon of the same initial wall thickness.

When looking at how fast Helium leaks out of a 260, as compared to round balloon, the overall rate is not equal. Consider that a round balloon has greater initial lift due to a higher volume : surface ratio, right? (the lift being due to the volume, the major weight being that of the “surface”). Where does the gas escape? Through the surface (the latex). Well, a 260 has greater surface : volume ratio, so it should have a faster relative deflation.

Generally, a helium filled 260 sculpture has “landed” and is starting to droop in just a few hours, whereas a round balloon may have an “float” time of 8 – 10 hrs., and have landed, but still round the next day. The difference is that in a Helium-filled round balloon there is so much more initial lift than in a 260 that even after the round balloon loses some Helium, there is enough lift left to keep it afloat. (enough lift left… enough lift left… say that 10 times as quickly as you can 🙂 Also, consider the *apparent* deflation. A round balloon can loose some volume without being visibly much smaller, whereas a 260 sculpture can begin to “droop” with minimal deflation. In a helium-filled 260, you only have a few tenths of a gram of lift, and that is quickly lost as soon as just a little Helium diffuses out of the balloon and the diameter starts to decrease.

I’m guessing that the bigger the balloon, the better it will float?

Yes – the more weight it will lift off the ground.

Note that at high altitudes in Utah and Colorado, some foil balloons won’t float; because of the low air pressure above 5500 feet, the buoyant effect of the helium is decreased.

Water vapor in the air (humidity) lowers the molecular weight of the air and decreases the buoyant effect of the helium. Mike Barr writes: Here in the New Orleans area we’re 3 ft below sea level and 85-90% humidity is normal.

  1. I think most balloon decorators do this from choice, but here you must only use helium and not balloon gas (a mixture of air and helium).
  2. We don’t sell (round) helium filled latex balloons without Hi-float, because without it, the float time isn’t long enough to satisfy anyone. However, we often can’t use as much, because of the humidity. In the New Orleans area, latex balloons with the same amount of Hi-float used in Denver won’t float (because of the added weight of Hi-float).
  3. Watch the outside temp carefully. A fully inflated balloon will pop 4 seconds out the door on a warm day.

All of my suggestions are based on experience, so do what satisfies your customers best and makes you the best living.

I wonder what the effect of high float in a 260 or a 350 would do if it were filled with helium. My guess is that if would make it too heavy to float, but I know it would seal the pores. So, I don’t know. Sounds like a good experiment to test and see what happens.

In *general*, I find that the more twists a 260 has, the faster it seems to deflate.

What sculptures can I make that will float?

You can fill a 260 with helium but there’s no guarantee that it will float, because the 260 will only hold a limited amount of helium and usually the weight of the balloon will prevent it from floating. A 350 will float without a problem, although it still won’t float for very long. If you want a helium 260 to float in mid-air you’ve got to fill it up all the way to overcome the weight and then there’s nothing to twist/no place for the helium to be pushed to. I know 260’s will float at sea level but higher elevations may be different. So you can fly sculptures, but only ones you can make out of full balloons, such as the heart, candycane, spiral, octopus, etc.

Try this: inflate a 260 and tie it end to end. (After you’ve tied the ends together, you can do an apple twist to hide the knot.) Then attach a string or ribbon and you’ve got a hollow balloon… similar to the look of a Geo only with a much bigger hole in the middle.

When I first started twisting I was in a store with a helium tank and I taught the old gentleman there how to make a basic dog. We then got several to float to the ceiling. They were just about completely filled.

I love floating sculptures. I do a variety of “floating” sculptures at various functions and have always gotten tremendous response from them. In addition to the traditional “heart on a string”, I do planes, hummingbirds, butterflies, helicopters, swans, hats that won’t stay put :), a 5′ tall floating Goonie Bird, flying fish, and a bunch of others. Let your imagination go!

I tie 260’s to round helium balloons, so just about anything will fly. Two 260’s attached to one 11″ balloon will hover (this is going to depend on how stingy they are with the gas at the store, however). I can adjust the balloon’s ability to hover by tying on additional ribbon, then snipping it off as the balloon starts to sink. This way, the kid gets to have the fun of a floating sculpture without it disappearing into the stratosphere. I specifically used this to float Larry’s bats last Halloween, and used them for party games.

One of my favorite things to attach to a hat is a hot air balloonist attached to a round. I make 1/2 of a guy out of a 130 and then I make his basket out of another 130 and attach them to the string of the round balloon and then tie the string to a hat.. It never fails to get a hoot.

A “huggy bear” on a helium balloon string has a kind of Whinnie-the-Pooh look.

What can I do with floating sculptures?

Our family does balloon launches on the 4th of July afternoon, instead of fireworks. The 260’s don’t have much volume so a small tank fills a lot of them. Just one sculpture at a time, and watch them float out of sight, listening to the cheers from the neighbor’s yards. The kids love it, and nobody gets burned. (We spend the 4th at a relatives house on a hillside overlooking where the city does the public fireworks display. It passes the afternoon hours, waiting for dark and the big show.) After dark, if you have a strong spotlight you can follow a balloon quite a ways, and let the neighbors claim a U.F.O. sighting.

I’ve also used a clear Geo Blossom as a prop on somewhat larger airplanes to give them increased lift. I attach a peice of string as to a round balloon. It’s fun to watch the kids flying in formation.

MB 7/6/96
MB 1/4/97

The following material has been saved from posts on the mailing lists. Rather than keeping it hidden away, it has been temporarily placed here until the guide editors get a chance to move it to its proper location in this chapter. Feel free to make use of it.

A 5″ round has almost neutral buoyancy in air when inflated with helium. They will be slightly lighter than air (they will rise) if overinflated. They will be slightly heavier than air (they will fall) if underinflated. Experiment to find a size (or combination of sizes) that provides the illusion you are looking for in your application. Note that you can trim off some of the nozzle to lighten balloons that fall.

If you need to prepare them ahead of time, and then use them later, you may want to take into account the length of time they will sit before the effect takes place, since they leak helium very fast. An overinflated – slightly lighter than air -helium filled- 5″ balloon will only float for an hour or so. make sure that whatever size you’ve inflated your 5″ to with helium that they will reach that “slightly lighter than air” stage before you need them. Always do an experiment first to investigate all the dimensions, including time. (In other words, “Kids, TRY this at home”)

I have the kids guess what I am making out of balloons, it might take 2 or 3 year olds a while to guess but that’s ok, all the guesses can give you ideas for new sculptures. Whoever guesses it wins it. If they win one then they can help other kids guess.

For a helium filled balloon to float in midair it would have to have neutral buoyancy. It is _very_ hard to achieve this condition – typically the balloon will either sink or float, but not stop half way. A simple figure made from a 350 MIGHT be able to float, or you could incorporate a round (at least 9-inch) balloon into the sculpture. Tie a LONG ribbon to each figure, and then add enough weight to the balloon so that it barely floats. As the balloon rises, it lifts more of the ribbon from the floor, until the weight of the ribbon equals the lift of the balloon, and a state of equilibrium is reached. Try to get equilibrium with about 6 feet of ribbon. Cut the ribbon off about a quarter inch above the floor at the beginning of your event. The balloons will then rise to the ceiling until they lose enough helium to equal that quarter inch of ribbon. Then they will begin very slowly descending until the ribbon touches the floor. After that they will sink very slowly to maitain the equilibrium between the balloon and the weight of the ribbon.

VALVES ON BALLOONS
some balloons sold at fairs come equipped with plastic self-sealing valves, so the average person can easily inflate the balloons without having to tie knots. Often, these valves have a string or ribbon attached to them for further time savings. These valves allow re-inflation. the valve will *reduce* floating time, as they don’t seal as well as a well-tied knot, and the extra weight of the valve will tend to pull the balloon down quicker!

HELIUM
We here in Aust. import ALL helium from North America. Would you believe that one major supplier ran out of helium today? So we wait for the next ship to arrive? Hope it isn’t too far away! Could have been worse I guess…. at least we had enough gas for the Xmas / New Year sales. Costs us anything from US$0.20c per 11″ balloon to $0.45c per balloon. Depending on discount level and supplier.

I am just curious. I do not want to start any fueds on the mailing list… so email me DIRECT and details will stay CONFIDENTIAL! I will post information of general interest without being specific to states or company names. The mailing list is not the place to quote specific prices or deals or supplier’s names. Example below. Is anyone interested?

I hear that helium in N.Z is almost double the Aussie cost. In some parts of SE Asia they are still puting Hydrogen in balloons. (Put out that cigarette)…… because it’s cheaper than helium? What about Britain and Europe? Does their helium come from the North Sea oil feilds, or from Poland? Either way, it should be relatively inexpensive. I’ve also heard that small amounts of helium are now coming through from Saudi. Can anyone confirm?

For what it’s worth… I beleive that the single biggest factor in determining a countries “balloon market potential” is the price the retailers must pay for helium. Seriously, helium balloons are only afforded by the upper income levels in some countries.

In Colorado, much like the price levels I was paying in Missouri, it was around 5-6 cents a balloon tops. Perhaps a little cheaper in Missouri. The biggest source that I’m aware of (outside of the natural helium deposits in and around Texas), is the waste helium that is stripped from Natural Gas during the refining process. I recall reading that we jettison about a TON of it a day. The only thing we seem to be paying for here is the storage, transfer and tank rental.

Seems that any country with Oil and Natural Gas should have plenty of Helium, and that would mean Saudi Arabia should be floating away in the stuff.

I’ve heard that Helium is expensive in Europe as well as most countries south/near the equator.

HEALTH RELATED
Inhaling Helium Can Cause More Than a Funny Voice
http://www.kidsource.com/kidsource/content2/news2/helium9_3_96.html

INDUSTRIAL
BOC Gasses Products Helium
http://www.boc.com/gases/products/helium/helium.htm

(“Helium is extracted from natural gas deposits. Only a few sources in the world contain a significant proportion of helium and justify its separation. These are in the US, Poland, Algeria and Russia. Because of its high value, helium is the only major industrial gas to be traded internationally”)

ELEMENTARY
Periodic Table of Elements
http://www.sci.ouc.bc.ca/chem/web-elements/He.html

Helium
http://www.crl.com/~peters/periodic/2.htm

helium is not flammable, but it is sold as a compressed gas which can become a hazard if handled improperly.

Helium is an element. After it is released into our atmosphere, it is so light that it floats off into outer space. Once current supplies trapped underground are exhausted, we’ll all be out of luck.

The only way I know to “manufacture” it is by the radioactive decay (“alpha particles” are Helium) of elements like Uranium, Thorium, Radium, Radon, Polonium, etc. Kids, don’t try this at home or the NRC will come knocking at your door.

Hmmmm, it’s also possible to produce Helium by fusion of Hydrogen isotopes – the way the sun does it – but it will be many, many years (if ever) before anyone can do that continuously and reliably here on earth.

Don’t forget that when you are comparing helium prices, purity matters. The purer the helium, the longer the float time. Some helium distributors are selling “balloon grade helium” — helium mixed with air. Unless you like using lots of “Hi-Float,” stick with the good stuff.

Perhaps It is only my supplier, and the fact that I heard this as well and chose to ask the tank fillers and sales personel. But the only difference between balloon grade and medical/ industrial grade is that they clean the med/ind tanks of any possible particulates or gas impurities during each fill both grades of helium are filled without mixing from the same tanker truck, which is treated at it’s source as though it were to be used only for med/ind use.

From my 13 years in the balloon industry, I have found that the change in pressure in a helium tank is almost proportional to the change in volume. So you can assume that when the pressure drops to half you have used up half your gas. (Any variation from this is to small to worry about).

Thanks to everone who contributed to this topic on “Helium – Cost and Origin”. At last count I had received over 30 direct replies from around the world. I promised not to embarras anyone by disclosing specific deals or mentioning suppliers by name or whereabouts. I hope everyone can accept that I have no intentions of fueling price wars or commercial bickering. If this information educates some of us a little…. and helps us understand the “commerce” of our product, then it could lead to future improvement in our industry. Knowledge breeds innovation! Make sure your gas rep gets to read this summary. His company may decide to get a little more aggressive in the marketplace.

____________________________________________

Answers to the TRIVIA questions:- “Air Liquide” (France) claim to be the world’s largest industrial gases company. “BOC Gases” (UK) lay claim to supplying more helium worldwide than any other industrial gases company. Source of Information:- Australian HQ for each company.

PRICES
Got to be real careful here…. I will assume currency mentioned is LOCAL currency, unless otherwise stated. You have to do the conversions. I am also ASSUMING that everyone that mentioned a COST PRICE has INCLUDED tank rental, delivery charges, sales tax (if applicable) etc. In Australia the COST per 11″ latex is from US$0.18cents to 45cents. There are 3 gas companies that import helium to Aust. All of it comes from North America by sea. In Canada the cost to fill an 11″ latex balloon with helium is marginally dearer than in the USA.(to be expected) Around 1.5 to 2 cents. My information is that one gas company supplies all other gas companies in certain states or provinces. (is cartel the correct term?) Helium source is Texas. This is also common practice in Australia between the gas companies in certain states. In Latvia (Scandanavian country) all helium is drawn from natural gas deposits in nearby Russia or Sweden.A cylinder that fills 600 x 9″ latex costs approx US$107 from Russian suppliers…. but US$148 from Swedish suppliers. In New Zealand and the Pacific Islands….. approx 1.5 x the Australian prices. Source of helium….. the USA.

In Britain they want to delay comment till after they change over to the new “Euro Currency”. ?????? The truth…..No email received.

In the USA…. the cost prices quoted to me ranged from 5cents per 11″ latex to near 20 cents per 11″ latex. I hope the 5cents was an exagerated brag… and the 20 cents was a mathematical error. The typical prices quoted were 6.5 to 9 cents. No, I won’t tell anyone who or where! I do not want to expand upon the above in detail – Please?

I don’t think I need to go into the “properties of helium” issue. That became a public topic on the list, and repeating it or challenging some of it is not necessary. The balloondeco archives now has a wealth of information on this topic. Knock yourself out!

BOC Gases has a web site. I quote from it;
“Helium is extracted from natural gas deposits. Only a few sources in the world contain significant proportion of helium and justify it’s separation. These are in the US, Poland, Algeria and Russia.Because of it’s high value, helium is the ONLY major industrial gas to be traded internationally” Thanks to Sheena Beaverson for the research.

The BEST literature I have seen recently on helium production and uses is from “Praxair”. The colour folder is called “Helium: Supply is Just The Beginning”.Absolutely Amazing! Pick it up and teach yourself (and your business) a thing or two about helium. Their web site is; http://www.praxair.com They have offices all over the world…. and No, we don’t buy helium from them!

GENERAL
Balloon Sculptures don’t work well submerged in water. I find the buoyant forces on balloons tend to break apart the sculptures if you try to submerge them.

3/17/99

A 6″ heart fully inflated with helium will obtain equilibrium. Meaning: wherever you put the balloon, it stays. I made a 6 balloon flower with 6″ hearts and the hearts (really inflated) would hold up the weight of a stem. It waved in the wind very nicely!!

A fully inflated 350 will rise vigorously. I haven’t tried it yet but I think I could make a cool zeppelin that will stay put. A fully inflated 5″ smiley will rise.

I have used helium in 9″ and 11″ rounds and made things like spiders and ghosts with the 260’s for legs (having air inflated the 260’s) and the round holds the sculpture up in the air for quite a while and it floated around the restaurant I was working. It kind of went with the air currents and seemed to have a life of it’s own. You can also attach rounds filled with helium with a string to 260 sculptures and have them hold the sculpture up.

We have done this also. Used a round helium filled balloon with fishing line to float spiders around a halloween party. It took a lot of experimentation to get it right. How much helium so they wouldn’t sit on the ceiling or drag on the floor. How big a spider sculpture etc. The end effect was great, but it was work to get it right.

Of course, you just could do what I do: Slap a piece of tape on the balloon when no one is looking, and say “Hey kids, betcha can’t do this. Here, everyone take a needle and try stabbing it through something.”

The only down-side to the tape “stick” is that this is a technique taught in boy-scout and cub-scout magic books. I’ve had more than one kid inform me that they learned how to do it in scouts so I let them inspect the balloon and when there is no tape you make major points

How to charge for He filled balloons
Since an 11″ helium balloon holds .5 cubic foot of helium, take the cost of your helium tank (include the rental and portage charges, too) and divide that cost by the cubic feet that it holds (there are many different sizes – so ask your supplier how much yours holds). Then divide that number by two and you have your cost per balloon.
Ex:
$30 (cost of tank) divided by 200 cu ft. = .15 cost per cu. foot. Divide by 2 to get cost for .5 cu ft. = .075 cents per 11″ balloon. Keep in mind that you will pop some balloons and lose helium, so be sure to add a little to the final price to cover that cost. Then
1) include the cost of the Hazardous Materials Charge.
2) include the cost of delivery if any.

An eleven inch Qualatex balloons weighs .108 oz.

I never use 9″ balloons for helium work because their float time is too short.

On page 19 of the Jan/Feb 1994 issue of Images there is a chart showing balloon lift ability and helium capacity.