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Compressed Air as an Energy Source in Pneumatics
The Composition of Air:
Air is a substance which can be found in unlimited quantities practically everywhere on the planet. Atmospheric air is a gaseous mixture primarily made up of two gases, namely 
78% nitrogen (N_{2}) by volume  
21% oxygen (O_{2}) by volume 
as well as traces of other gases like e.g. carbon dioxide, helium or argon (see the pie chart above). Furthermore, depending on the climatic conditions, air also contains a certain percentage of condensed water (humidity) as well as impurities like dust, soot and other solid particles. The basic prerequisite needed for the smooth operation of pneumatic systems is compressed air that is both dry and clean.
The State Variable Pressure:
Air that is compressed for the purpose of industrial applications is referred to as compressed air (nowadays the antiquated term pressurised air which was sometimes in use is rarely found). Besides the temperature, the pressure p is one of the most important state variables used in physics. It is defined as a force F, which is distributed evenly in all directions over the surface A
By way of the unit of force, the newton (N), we can directly derive the unit of pressure, the pascal (Pa):
A unit of pressure that matches the order of magnitude of atmospheric pressure is represented by the physical unit, the bar:
1 bar = 100,000 Pa = 0.1 MPa
1 mbar = 100 Pa = 0.001 bar
The various measurements of pressure are distinguished from each other only in terms of their reference point. The most precise reference point is the pressure of an absolute vacuum. The following table provides an overview.
Absolute pressure  (p_{abs})  Pressure which is related to the zero pressure state existing in an absolute vacuum. 
Atmospheric air pressure  (p_{atm})  Subject to weatherrelated fluctuations (at sea level it averages 1013.25 mbars, for rough estimates it can be rounded off to p_{amb} = 1 bar.) 
Overpressure  (p_{o})  The absolute pressure is greater than air pressure. 
Underpressure  (p_{u})  The absolute pressure is lower than air pressure. 
Pressure with respect to p_{amb}  (p_{e})  Pressure above or below atmospheric pressure 
In pneumatics the pressure specifications are generally related to atmospheric pressure. A pressure of p_{e} = 1.5 bars is thus a pressure, which is 1.5 bars above atmospheric pressure. The subsequent diagram provides another graphic depiction of these relationships.
Compression and Expansion of Air:
Air is compressible, i.e. it can be pressed into a more compact space. During this process volume V, pressure p and temperatur T are interrelated in a rigid and coherent fashion:
As the temperature increases the pressure rises, as long as the volume remains constant (example steam pressure building up in a kitchen pressure cooker) 

If the volume is reduced while the temperature is kept constant, the pressure increases (example: slowly pumping using a bicycle pump) 
This qualitative relationship can be reproduced mathematically in the formula
i.e. for two different states (subscript 1 or 2) it holds that
The following animation illustrates this context.
If the temperature remains constant while the state varies (i.e. T_{1} = T_{2}), then the change of state is termed isothermal and the relationship can be expressed in a simplified form using the BoyleMariotte law
A simple mathmatical example should help to illustrate the application of this law: air with an initial volume of V_{1} = 2 m^{3} and an initial pressure of p_{1abs} = 1 bar is compressed at a constant temperature to a volume of V_{2} = 1 m^{3}. How high is the pressure p_{2abs} in its final state?
To solve this problem we invert the BoyleMariotte law for p_{2abs} and insert the given numerical values. Thus we obtain
Therefore we can see that by halving the volume the air pressure has just about doubled. It remains to be noted that in actual technical applications the assumption of constant temperature during compression or decompression can normally only be applied approximately, due to the fact that the change in volume cannot occur a sufficiently slow rate to keep the temperature at a precisely constant level.