Question

You place an online order for 13 perfectly round Latex Crystal-tone Color Balloons (diameter of 1.000 ft) which completely inflate at a pressure of 1.5 atm, 2 packages of Quickie Balloon Clips (they only come in packages of 12), and a roll of White Curling Ribbon (you opt for the special deal of the 500-yd roll). To enhance your savings, 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: 2.125 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 will make your wallet happy?

Extra info:

(A) Room temp = 25 degrees Celcius

(B) Helium may be considered ideal only at low pressures

(C) It is relatively easy to solve the Van der Waals equation for pressure, solving it for volume or number of moles leads to cubic equations.

(D) Van der Waals constants for He: a = 0.03502 L^2 atm mol^-2, b = 0.0237 L/mol

Answer 1

The ballons being completely round and 1 ft (30.48 cm) in diameter the total volume of each balloon is 14.8 L. Radius is 15.24 cm.

13 balloons will require 14.8L x 13 = 192.4 L of gas at 1.5 atm for completely inflating them.

PV = nRT

1.5 x 192.4 L = n x 0.082057L atm K⁻¹ mol⁻¹ x 298 K

n = (1.5 x 192.4)/0.082057 x 298

n = 11.80 moles is the amount of helium gas required.

four sizes of helium gas cartridges available:

2.125 L at 120 atm, 2.500 L at 125 atm, 3.000 L at 110 atm, and 3.250 L at 100 atm.

The number of moles in each of them can be calculated from the Van der Waals equation

I have computed these values online

In the Gas cartridge 2.125 L at 120 atm the number of moles is n = 9.34 moles

In the Gas cartridge 2.5 L at 125 atm the number of moles is n = 11.39 moles

In the Gas cartridge 3.000 L at 110 atm the number of moles is n = 12.19 moles

In the Gas cartridge 3.250 L at 100 atm the number of moles is n = 12.11 moles

Since we need 11.80 moles the correct gas cartridge that would be cost effective would be 3.250 L at 100 atm