It is logical to detect from the general gas equation (formula 2.3.1.1) that heat must, in principle, be generated when air is being compressed. In order to arrive at a theoretical calculation of the quantity of heat, one looks at the simplified case at which compression takes place theoretically without any heat being added or withdrawn.
Formula 2.4.1 As part of the heat of compression is conducted away from every compression chamber, compression in practice proceeds in between adiabatic2 and isothermal3, i.e. in a polytropic4 mode. The following interrelations can be deduced from the gas laws: Temperature change as a function of pressure at constant volume as isochores5
Formula 2.4.2
The volume change as a function of temperature at constant pressure as isobars6. 
Formula 2.4.3
Pressure change as a function of volume at constant temperature as isotherms
| 2 Exponent x | 4 Exponent n > x | 6 Exponent n = 0 |
| 3 Exponent n = 1 | 5 Exponent n = |
Formula 2.4.4
Supplementing the formulae indicated above, which are more of theoretical interest, there are the following equations which have greater practical importance. The adiabatic pressure change as a function of temperature at constant volume,
Formula 2.4.5
respectively the adiabatic volume change as a function of pressure at constant temperature
Formula 2.4.6
Polytropic change of pressure as a function of temperature at constant volume, 
Formula 2.4.7
Respectively the polytropic change of volume as a function of pressure at constant temperature,
Formula 2.4.8
The change of state of a gas is established by taking into account the relationship of specific heat capacities exponent x. In doing so, one differentiates between:
cp = specific heat at constant pressure respectively referred to 1 kg (kcal/kg°C)
cv = specific heat at constant volume respectively referred to 1 kg (kcal/kg°C)
Based on experiments, the relationship of specific heats x = cp/cv is:
with uniatomic gases x = 1.666 = 5/3
with biatomic gases x = 1.400 = 7/5
with triatomic gases x = 1.333 = 8/6
The following illustration shows the interpretations of the various specific cases of the general polytropic law.
Figure 2.4.1
The difference of the specific heat referred to 1 m3 at constant pressure and volume is a constant. For ideal gases, the following applies when applying the above x values:
With uniatomic gases
| | 5 | | | |
| cp = |
| = 0.22 kcal/m3°C | | cV = 0.13 kcal/m3°C |
| | 22.4 | | | |
with biatomic gases
| | 7 | | | |
| cp = |
| = 0.31 kcal/m3°C | | cV = 0.22 kcal/m3°C |
| | 22.4 | | | |
with triatomic gases
| | 8 | | | |
| cp = |
| = 0.36 kcal/m3°C | | cV = 0.27 kcal/m3°C |
| | 22.4 | | | |
The specific heats of real gases deviate all the more from those of ideal gases, the higher their number of atoms. Furthermore, the specific heats, compared with the behaviour of ideal gases, increase with temperature and with pressure. For calculations involving a larger range of temperatures, the median specific heat cm must, therefore, be used instead of the true specific heat.
Table 2.4.1| Pressure bar absolute | Temperature °C |
|---|
| 0 | 50 | 100 | 200 | 300 |
|---|
| 1 | 0,2402 | 0,2409 | 0,2418 | 0,245 | 0,250 |
| 50 | 0,264 | 0,256 | 0,254 | 0,252 | 0,253 |
| 100 | 0,288 | 0,270 | 0,263 | 0,257 | 0,257 |
| 150 | 0,306 | 0,282 | 0,271 | 0,262 | 0,261 |
| 200 | 0,322 | 0,292 | 0,278 | 0,267 | 0,265 |
| 250 | - | 0,299 | 0,285 | 0,272 | 0,269 |
| 300 | - | 0,303 | - | - | - |
