An isentropic expansion or compression of an ideal gas follows PV^k=constant, where k=Cp/Cv. Through the ideal gas equation (PV=nRT) we can also get PT^((k-1)/k). This may be what you're thinking of. Of course, this only applies to ideal gases. Real expansions and compressions aren't isentropic. If k=1 then you're not going to get any cooling effect by expansion because (k-1)/k=0. That's why we prefer refrigerants with k as large as possible, like ammonia. We must also consider latent heat because required mass flow = cooling load / latent heat. The more fluid you have to pump around to get the job done, the more losses you will have and the more equipment expense too.

Thanks, Dudley J Benton for the answer, does the high value of k also affects operating parameters such as compression ratio or discharge temperature, etc., and performance (capacity and power consumption) of the system?

Let's say you wanted to design a refrigeration system to keep the Pitzer vaccine at -25°C. You want this device to operate in the storage room of a hospital that is kept at +25°C. Traditionally, you would open the ASHRAE Handbook of Fundamentals, turn to the section containing a list of refrigerants sorted by optimum operating temperature, and pick the best one. The three logical choices would be: R502, R504, or R13b1. You would then choose one of these for various reasons, including: availability, cost, compatibility with lubricants and seal materials. Let's say you pick R13b1. The optimal cycle temperatures would be perhaps -50°C and +50°C, providing a ⌂T of 25°C on both sides. The operating pressures are fixed by these two saturation temperatures at: 144 kPa and 2750 kPa. You would then select a compressor that could handle these and has a compression ratio of at least 2750/144=19:1. What I'm saying is that you don't start with k. You start with the requirements, then the available fluids, then the equipment (compressor, heat exchangers, seals, etc.). I have attached a typical table from which you select a refrigerant for your specific purpose. It's on the "cycle" tab in the "refrigerants.xls" spreadsheet that comes with the property AddIn at this location http://dudleybenton.altervista.org/software/Refrig100.zip