Assume the helium and air to be at 0°C and 1 atm.

Question

A modern-day zeppelin holds 9,280 m³ of helium. Compute its maximum payload at sea level. (Assume the helium and air to be at 0°C and 1 atm.)

Answer 1

Given:

Volume of Helium = 9,280 m³ 

Air temperature = 0°C (273° K)

Air pressure = 1 atm.)

Gravity's acceleration g = 9.8 m/s²

Calculation:

STEP 1: Helium Density

We need to find the densities of Helium and Air at T = 273° K and P = 1 atm.

From the Periodic Table of elements, Helium has molar mass of 4.0026 g/mol. Thus, from the gas law:

Density of Helium = Air Pressure x Molar mass / (Absolute Temperature x Gas Constant)

........................- P x M / (R x T )

........................- P x M / (R x T )

Where

P = 1 atm

R = 0.082057 ( L, atm / mol .K)

M = 4.0026 g/mol.

T = 273° K

Then

densities of Helium =

( 1 atm x 4.0026 g/mol) / ( 0.082057 ( L, atm / mol .K) x 273° K)

= 0.1785 g / L

= 0.1785 kg / m³

STEP 2: Air Density

Air is approximately made of 80% Nitrogen, 20% Oxygen, but water and other contents affect air density. We will used an air-density calculator to find this constant as

Air density at pressure 1 atm and temperature 273° K ≈ 1.28 kg/m³

STEP 3: Air Lifting Force (Buoyant Force)

The Helium balloon of Volume = 9,280 m³ displaces equal volume of air. Thus, the displaced air exerts force = mass x acceleration.

The mass of the displaced air by the Helium balloon = Volume of Balloon x density of air

=  9,280 m³ x 1.28 kg/m³

When multiplied by g = 9.8 m/s², we get the air-lifting force:

= 9,280 m³ x 1.28 kg/m³ x 9.8 m/s²

= 116408.N

= 116.408 kN. ------->> Air Buoyancy

STEP 4: Weight of Helium Balloon

This is the downward weight of the balloon that resists buoyancy.

The weight of the Helium balloon = Volume x density of Helium x g

= 9,280 m³ x 0.1785 kg / m³ x 9.8 m/s²

= 16233.504 N

= 16.233 kN

STEP 5: Maximum Payload

The maximum payload of the Helium Balloon is the difference between the upward air lift (buoyancy) and downward weight (drag);

Payload ( in Newton) = 116.408 kN - 16.233 kN

...................................≈ 100.175 kN

In terms of kilograms, we divide the Newtons by g to get

Payload (in kg) ≈ 100.175 kN x 1000 / 9.8 m/s²

....................... ≈ 10222 kg

.........................≈ 10 ton