Adiabatic compression by mechanical means is the most common model for compression of the martian atmosphere for in-situ resources utilization. There are other methods, such as thermal adsorption but they are generally less efficient.
Adiabatic Compression power equation
When the compression is small, we can get away with the following simple equation:
W= compressor power (kW)
dp=pressure change (kPa)
n=compressor efficiency (usually between 0.6 and 0.8)
But when the compression becomes significant, we need to take into account the work done in compressing the gas. The equation for adiabatic compression (no heat loss at the compressor) is still not exact, but it is OK for settlement design.
W= power (kW),
y= Heat capacity ratio of the working fluid = 1.67 for noble gases = 1.41 for other gases, including air
mf= mass flow (kg/s)
R= Ideal gas constant = 8.314 J/kg mole
T= Absolute gas temperature before the compressor (K)
w= Molecular weight of the gas (g/mole)
n= Efficiency (from 0,null to 1, perfect)
P2= Absolute gas pressure after the compressor (kPa)
P1= Absolute gas pressure before the compressor(kPa)
Table 1, gas molecular weight, many gases are diatomic in their natural state.
The following link provides an Excel spreadsheet for compression calculations, and an example for the compression of the martian atmosphere.