Question

Your mother put you in charge of your little sisters birthday party who happens to turn 13 years old on the very day this homework assignment is due. Being buried in work and trying to keep up with preparing for lectures, completing homework assignments, and cramming for upcoming exams, you would like to maximize your study efforts and not worry about the party. Still, you don't want to offend your mother, particularly because she keeps reminding you every 3 hours. Finally, you give in and decide to surprise her with 13 helium filled balloons. You place an order for 13 perfectly round latex crystal tone color balloons (diameter 1.00 ft) which completely inflate at a pressure of 1.5 arm, 2 packages of balloon clips (they only come in packs of 12), and a roll of white curling ribbon (you opt for the special deal of the 500 yd roll even though you will only need 20 yd). To enhance your savings from the online order even more, you use your knowledge about gas laws to figure out how much helium you would actually need. There are four sizes of helium gas cartridges available: 2.215 L at 120 atm, 2.500 L at 125 atm, 3.000 L at 110 atm, and 3.250 L at 100 atm. The price is calculated according to pressure times volume.

Which one is the best choice making your mother, your sister, and your wallet happy? Please explain.

Answer 1

Let us the assume the balloon be in the shape of sphere

Volume of the sphere =

Given diameter of each ballon = 1 ft = 0.3048 m

Radius of the balloon = 0.1524 m

Volume of the sphere (balloon) =

Volume of the ballloon =

Volume of the balloon = 0.01482 *1000 L = 14.8 L

Volume of 13 balloons = 192.6 L

Pressure in the balloon = 1.5 atm

Pressure in the 13 balloons = 19.5 atm

Pressure * volume = 192.6 * 19.5 = 3756.6 L atm

But we have 4 sizes of helium catridges

Let us calculate P*V for each catridge =

first catridge =2.215 * 120 = 265.8 L. atm

second catridge = 2.5 * 125 = 312.5 L.atm

third catridge =3.0 * 110 = 330 L.atm

Fourth catridge = 3.25 * 100 atm = 325 L.atm

Now let us find how many catrideges of each type do we need

For this

Divide 3756.6 L.atm by 265.8 L.atm

Similary for divide 3756.6 by all catrideges

Among all 11.4 is minimum.

So it is better to buy catridge with 3.0 L at 110 atm about 11.4 numbers.