# How to calculate impulse from pressured gas?

Let's say we have a 1L bottle filled with compressed air at 10 atm.

If the bottle is opened, how can we calculate the impulse applied by the releasing gas?

I'm guessing the diameter of the opening doesn't matter very much, is that correct?

Part 1) For a short time $$\delta t$$

The impulse is Force x time

$$F=PA$$

$$A$$ is the area of the opening - so the impulse is $$PA\delta t$$

Part 2) If all the gas is released.

The total impulse could be estimated if we assume all the atoms of gas (of mass $$m$$) leave the opening in the same direction, with velocity $$v$$, determined by the temperature ($$\frac{1}{2}mv^2 = \frac{3}{2}kT$$), where $$k$$ is Boltzmann's constant.

The impulse from each atom is $$mv$$. If there are $$N$$ atoms and total mass of gas $$M$$, the total impulse is $$N\times mv = Mv$$, with $$v$$ being determined from the equation above.

For this particular question you could link some of the quantities involved with the equation of state formula $$PV=nRT$$, but you would still need information about the atomic number of the gas and either its temperature or the total mass.

• Thank you for this detailed answer. May I ask about the formula in part 2 linking velocity to temperature; what is this equation called? It looks like it's saying kinetic energy is equal to a constant of temperature, but I don't understand where the pressure inside the container fits in. Or is this v the velocity of each individual atom doing its Brownian motion? Oct 15, 2021 at 19:50
• @ CaptainCodeman Yes, $v$ is the velocity of each individual atom. The formula, that you'll find is $K.E = \frac{3}{2}kT$ Hope it all makes sense Oct 15, 2021 at 23:56