Pressure of sealed in liquid nitrogen

Question

If you

1. take a container of great strength,
2. fill it to the brim with liquid nitrogen,
3. seal the container, and
4. heat it to room temperature.

What will the pressure be inside?

Bonus: The same with liquid Helium.

(I posted this Reference for phase diagrams of elements , but the answer here can not be read from the phase diagram.)

Answer 1

From the densities of liquid nitrogen and nitrogen gas at standard pressure the volume ratio is about 1:700.

For an ideal gas in a closed 1L container this would result in a pressure of 700 atm according to $$P V = n R T$$ From the phase diagram nitrogen is a gas at standard pressure and becomes supercritical at approximately 100 atm. The ideal gas law can therefore only be a guideline well below this pressure. To incorporate the intramolecular forces the van der Waals equation is the next best choice: $$\left(p + \frac{n^2 a}{V^2}\right)\left(V-nb\right) = nRT$$ With the coefficient $b$ for nitrogen ($3.85 \cdot 10^{-5}$ m$^3$/mole from here) the minimum volume for 1L of liquid nitrogen (31 mole) is 1.2L. So even the correction to the ideal gas law cannot capture the full range.

Only the experiment might tell us what the real pressure might be. Someone has even done it inadvertently (accident report):

This vessel exploded at a pressure of 1200 psi or 82 atm, so the pressure of 1L of liquid nitrogen heated up to room temperature should be between 700 and 82 atm.