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This question got me wondering about the pressure inside a bombardier beetle.

Lots of articles mention pressure, but don't specify the amount of it:

One study records the velocity of the spray to be within a range of 325 to a stunning 1950 cm/s. [...] Once the muscles around the reservoir squeeze the first amount of reactants through the valve into the reaction chamber, the resulting explosion causes the pressure to rise rapidly in the reaction chamber, forcing shut the one-way valve. The products of the reaction then exit the chamber with a pop and a puff, and the pressure inside the reaction chamber lowers again, falling below the pressure of the collection reservoir, which is still being squeezed by the reservoir muscles.

Calculating from the spray characteristics is beyond me.

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According to this study, model data shows a maximum pressure of 110 kPa, that's 16 psi and only 1.086 atm:

Model data shows a maximum pressure of 110 kPa

The three-phase process involved in the beetles explosive secretory discharge (ESD) process. The inlet size is shown as a proportion of the inlet radius. During the first phase of refill and heating (blue), only the inlet valve is open. During the second phase (red), the chamber is closed. Finally, the third phase (marked by an arrow) is the exhaust phase, when only the outlet valve is open. This phase is significantly shorter than the first two phases. This cycle then repeats with phase 1 starting at approximately 5.1 and again at 7.8 ms.

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