Energy and Human Ambitions on a Finite Planet, 2021a

B.4 Ideal Gas Law 382
Example B.4.2 Gas is stored at high pressure at room temperature
in a metal cylinder, at a pressure of about 200 atmospheres. 23 The
cylinder is designed to meet a safety factor of 2, meaning that it likely
will not fail until pressure reaches 400 atmospheres. If a fire breaks
out <strong>and</strong> the cylinder heats up, the pressure will rise. How hot must the
gas get before the cylinder may no longer be able to hold the pressure
(assuming no fire damage to the cylinder itself)?
We could start throwing numbers into the ideal gas law, but we don’t
know the volume or number of moles (or particles). Heck, we’re not
even given a temperature. Ack! Students hate this sort of problem,
because it does not appear to be algorithmic in nature. No plug <strong>and</strong>
chug (an activity that does not engage the brain heavily, <strong>and</strong> thus its
appeal).
But we’re okay. What is room temperature? Something like 20–25 ◦ C,
so that’s 293–298 K. Whatever the volume is, or the amount of gas
in the cylinder, those things don’t change as the temperature rises. 24
What we’re left with is a straightforward scaling between temperature
<strong>and</strong> pressure (because the numerical factors are all constant for our
problem). Therefore, if temperature doubles, pressure doubles. 25
Hey, it’s doubling pressure that we are interested in, which will
happen if the temperature doubles. So if the temperature goes up
to about 600 K, we may be in trouble. It is easy to imagine that a
fire could create such conditions. Notice that we are not bothering
to say 586–596 K, but just said about 600 K. Do you want a precise
temperature when the thing will rupture? Good luck. The point at
which it explodes may be 405 atmospheres or it may hold on until
453. Also, how likely is it that all the gas throughout the cylinder is
at exactly the same temperature when being heated by a nearby fire?
So let’s give ourselves a break <strong>and</strong> not pretend we’re totally dialed in.
There’s a fire, after all.
23: . . . means 200 times atmospheric pressure
This is an example where internalizing the
ideal gas law for what it means, or what it
says is more important than treating it like
a recipe for cranking out problems. Don’t
just treat equations as mechanical objects:
learn what it is they have to say!
24: The gas is not leaking out, <strong>and</strong> the
cylinder does not change size—at least not
significantly—as it warms.
25: That’s one of the things Eq. B.4 is trying
to say, beneath all the bluster.
© 2021 T. W. Murphy, Jr.; Creative Commons Attribution-NonCommercial 4.0 International Lic.;
Freely available at: https://escholarship.org/uc/energy_ambitions.