Well we all are familiar that nitrogen is used to achieve low temperature.
I was just wondering how can one determine the temprature that could be achieved when phase change of nitrogen is done from gas to liquid state.
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You need to consult a phase diagram for nitrogen to answer this. Yes, it could be done, but nitrogen is a terrible refrigerant, and you'll have trouble finding somewhere to discharge heat to at the extreme low temps.
https://www.engineeringtoolbox.com/nitrogen-d_1421.html shows the properties. In the graph, you want the area where pressure and temperature affect the boiling point, or about 0-33 bar, and about -210 to -145 degrees C.
To get a refrigeration cycle (a heat pump is just refrigeration of the outside) to work, you need to be able to take heat from a cold place and move it to a warm place. In this case, pressurizing the nitrogen at -200 C and pressurizing it to about 30 bar so that you can cool it off to drop the temperature/pressure. Heat in a condenser travels from hot to cold, so since the temp is -200 C, where are you going to find somewhere colder? The next problem is the range of pressure versus the range of temperature. You have 30 bar pressure change versus about 35 degrees of temperature. That isn't much of a swing, so your performance will be quite low.
In practical terms, it's a useless endeavor.