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The temperature at which $\ce{NO_x}$ is in equilibrium with its liquid and vapor phase at 1 atm is about –84 °C. Does that mean $\ce{NO_x}$ exists as a liquid at that temperature? What about it's vapor phase then, because it's vapor and liquid phase should be at equilibrium, right?

Also, it's that the reason why compressed $\ce{NO_x}$ when released from its closed and pressurized container (e.g. 50 bar) at 20 °C becomes a gas while its temperature drops? Therefore, the compressed $\ce{NO_x}$ was in it's liquid state at 50 bar, right?

If that's the case, then what is the relationship between the vapor pressure curve and the liquid-vapor dome? How would you use the saturated liquid-vapor tables to predict the vapor pressure for your substance? And which specific volume would you chose to define your substance: its liquid specific volume or its vapor specific volume?

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The vapor pressure curve describes the relationship between temperature and pressure for which the pure substance can exist as liquid and vapor, together in equilibrium. So it is not necessarily all liquid or all vapor. Both can be present in different proportions. As far as specific volume is concerned, your tables should list both the vapor specific volume and the liquid specific volume. The specific volume of the combination of vapor and liquid is proportional to the mass fraction of each. (So you need to know the mass fractions of vapor and liquid).

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Every gas at decompressing cools down (adiabatic expansion).

See for example: https://physics.stackexchange.com/questions/14140/why-does-the-gas-get-cold-when-i-spray-it

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