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Balloons on Mars

Air and fluids · level 5, lesson 6 of 6

A question to start

A helium balloon 2.6 m across can lift 10 kg on Earth. How big would one need to be to lift the same 10 kg on Mars?

The idea

A balloon floats in air by Archimedes' principle. With volume \(V\), the air pushed aside has mass \(\rho_\text{air}V\), and the gas inside has mass \(\rho_\text{gas}V\). The mass it can lift (the envelope plus the load) is

\[m = (\rho_\text{air} - \rho_\text{gas})\,V.\]

Weight and upthrust both scale with \(g\), so \(g\) cancels here too. What matters is the density of the air, and Mars air is very thin.

At the same pressure and temperature, a gas's density is in proportion to the mass of its particles (its molar mass). Helium is 4.00 g/mol; Earth air 28.97; Mars air 43.49.

Lifting with helium Mars Earth
Air density 0.016 kg/m³ 1.217 kg/m³
Helium at the same pressure and temperature 0.0015 kg/m³ 0.17 kg/m³
Mass lifted per m³ 0.0145 kg 1.05 kg
Volume to lift 10 kg 690 m³ 9.5 m³
Width of a round balloon 11 m 2.6 m
Earth: 2.6 mMars: 11 m10 kg10 kg

Worked examples

1. Helium on Mars. \(0.016 \times 4.00/43.49 = 0.00147\) kg/m³. Lift per m³: \(0.016 - 0.0015 = 0.0145\) kg.

2. The volume. \(10/0.0145 = 690\) m³. A sphere: \(V = \tfrac{4}{3}\pi r^3\), so \(r = (3 \times 690/4\pi)^{1/3} = 5.5\) m, 11 m across.

3. Earth. Helium: \(1.217 \times 4.00/28.97 = 0.168\) kg/m³. Lift per m³: 1.05 kg. Volume \(10/1.05 = 9.5\) m³, \(r = 1.3\) m.

Try it

  1. Why does \(g\) cancel when you work out the mass a balloon can lift?
  2. How many times more volume does a Mars balloon need than an Earth one, for the same load?
  3. Hydrogen is 2.02 g/mol. Find its density at Mars's surface, and the lift per m³. Is it much better than helium?
  4. A Mars balloon must lift a 2 kg envelope and a 3 kg instrument. What volume of helium does it need?
Answers
  1. Both the weight and the upthrust are mass times \(g\).
  2. \(690/9.5 \approx 73\) times.
  3. \(0.016 \times 2.02/43.49 = 0.00074\) kg/m³; lift \(0.0153\) kg per m³. Only about 5% better.
  4. \(5/0.0145 \approx 345\) m³.

Common mistake

"Lighter gravity makes ballooning easy." The balloon weighs less, but the air's push is smaller in the same proportion. The thin air is what makes Mars ballooning hard.

Where it leads