Adiabatic expansion of compressed air produces much less work than isothermal expansion starting from the same conditions. This is obvious from the PV graph, and the fact that the downward slope of an adiabat is greater than that of an isotherm.
Adiabatic expansion of air Edit
During adiabatic expansion of air (a diatomic gas with extra degrees of freedom for rotation and bond flexing),
or solving for V as a function of P,
where is a constant which depends on the starting pressure and temperature and the amount of gas present. For a monatomic ideal gas, the exponent is 5/3 instead of 7/5, which applies to air. See wikipedia:Adiabatic process#Ideal gas.
Determining the value of Edit
The value of is constant along the adiabat, and not elsewhere. The temperature is constant along the isotherm. We must use the values of and at the point where the two curves cross to determine the value of .
Integrating to calculate the available work Edit
Using negative values to denote work done by a gas and positive values to denote work done on a gas, the work done by air which expands adiabatically from to is
Evaluating (with double check),
which applies to the starting point where the isotherm and the adiabat intersect, with and . Thus, substituting for , we get
Isothermal expansion of air Edit
The work available from air which expands isothermally is higher:
Comparison of the two Edit
The ratio is
which seems to be always greater than one. There is an error in this somewhere still.
units version 1.80 with readline, units database in /usr/share/units.dat 2084 units, 71 prefixes, 32 nonlinear units (1/10)^(2/7): Definition: 0.51794747 log(1/10): Definition: -1 ln(1/10): Definition: -2.3025851 (1/10)^(2/7)+-1: Definition: -0.48205253 7/2((1/10)^(2/7)+-1): Definition: -7.2606195 7/2((1/10)^(2/7)+-1)/ln(1/10): Definition: 3.153247 7/2((1/100)^(2/7)+-1)/ln(1/100): Definition: 1.0386548 7/2((1/2)^(2/7)+-1)/ln(1/2): Definition: 28.104765 7/2(((1/2)^(2/7))+(-1))/ln(1/2): Definition: 28.104765 7/2(((1/2)^(2/7))+(-1)): Definition: -19.480739 ln(1/2): Definition: -0.69314718 ln(300): Definition: 5.7037825