Can I use P1/P2=(T1/T2)y/y-1 to find the temperature when we pressure liquid refrigerant, for example liquid propane?

As this formula applies to perfect gases, and liquids aren't gases, you can't.

Anyway, don't expect the temperature of a liquid to change much as you pressure it, since its volume doesn't change much at usual technological pressures.

The volume compressibility of liquid propane is probably near 2GPa, so if you compress it to 300 bar (30MPa) it will loose only 1.5% volume, and receive a mechanical work of less than 30e6*0.015=450kJ/m3. Taking a density of 700kg/m3 and a specific heat of 1kJ/kg/K gives us a temperature increase of less than 0.6K: little.

This estimation is very rough but should suffice to tell that temperature doesn't change when compressing a liquid.

Enthalpy is correct. From what I recall, correct me if I am wrong, you can only treat low density high temperature liquids like a gas. I am also wondering if this question has something to do with not pressurizing the liquid but depressurizing the liquid by throttling since he refers to a refrigerant?

Regards,
Caleb

If you use a boiling water bath to vaporize the liquid, yes you can use the ideal gas laws.