Perpetual Mobile with compliance of the law of conservation of energy

[sm0ky2]

 exponential pressure decay with increasing distance from the ground.
the following was copied from some random college prep webpage

We are actually living near the bottom of an ocean of air.

At sea level, the weight of the air presses on us with a pressure of approximately 14.7 lbs/in2.   

At higher altitudes, less air means less weight and less pressure.  Pressure and density of air decreases with increasing elevation.   

Pressure varies smoothly from the earth's surface to the top of the mesosphere.  This table compiled by NASA gives a rough idea of air pressure at various altitudes (as a fraction of one atmosphere).
 


fraction of 1 atm average altitude
                     (m)                 (ft)
1                     0                    0
1/2               5,486.3        18,000
1/3               8,375.8        27,480
1/10            16,131.9       52,926
1/100          30,900.9       101,381
1/1000        48,467.2       159,013
1/10000      69,463.6       227,899
1/100000    96,281.6       283,076

[abv]

you're still missing it.....

It takes MORE ENERGY to expand it at lower altitudes.
the DIFFERENCE in ENERGY is the SAME as the BOUYANT WORK FUNCTION!!!!!!

there is no "extra energy".....  the work done to lift the load was already consumed by the expansion process
PRIOR to the lifting.....

I'm not sure what law of conservation of energy can be used between isolated systems.
The first isolated system is balloon with gas. This system knows nothing about altitude. System has efficiency 100%. Energy from gas decompression equal to gas compression.

Please imagine this situation.
For better understanding energy of conservation in first closed system, let's say this system has turbine with alternator and battery to store energy as electricity and electrical compressor for gas compression process. All of this devices and energy transformation process have efficiency 100%.
The balloon with compressed gas and load stay on the ground. In first closed system, gas decompression process is starting. Rotating turbine with alternator generate electricity and store into battery. This stored electrical energy from battery will be used for gas compression process in a future.
Next.
In second closed system, when buoyancy force will be more than force of gravity of balloon with load then balloon starts flying.
Again, the first closed system knows nothing about altitude. In this case energy wat was stored into battery still there, and no matter what height balloon reached.
Next.
When balloon reached certain height then In first closed system using electrical energy from battery the compressor will start process of gas compression. It will use all electrical energy to compress gas to same balloon volume. The first system is closed and has efficiency 100%.
Plus extra potential energy of load from certain height.

[sm0ky2]

I'm not sure what law of conservation of energy can be used between isolated systems.
The first isolated system is balloon with gas. This system knows nothing about altitude. System has efficiency 100%. Energy from gas decompression equal to gas compression.

It really does not matter which of the four conservative "laws" you choose to use to analyze these two system.
the first, second, third, or the zeroth.
you cannot isolate them from a 3rd conservative system, which includes altitude and atmospheric pressure.
this "100% efficient" adiabatic compression and expansion isolated system has air pressure around it.

You did not include the work function of the atmospheric air pressure. This is a derivative function of gravity and specific mass of the gas molecules, density, and temperature. It can be directly related to the Buoyancy Function of the second system, through an almost identical gravitational derivative.

The two work functions of these separate, isolated systems, are one and the same.
a derivation of the gravitational force.

Please imagine this situation.
For better understanding energy of conservation in first closed system, let's say this system has turbine with alternator and battery to store energy as electricity and electrical compressor for gas compression process. All of this devices and energy transformation process have efficiency 100%.
The balloon with compressed gas and load stay on the ground. In first closed system, gas decompression process is starting. Rotating turbine with alternator generate electricity and store into battery. This stored electrical energy from battery will be used for gas compression process in a future.
Next.
In second closed system, when buoyancy force will be more than force of gravity of balloon with load then balloon starts flying.
Again, the first closed system knows nothing about altitude. In this case energy wat was stored into battery still there, and no matter what height balloon reached.

----> At this point is where energy is not accounted for in your design.
In order to expand the balloon, the air pressure must do work against the atmosphere.
This is evident by the pressure still remaining in the first system after "decompression".
It will balance itself to the atmospheric pressure
In this way, the first system "knows" something about altitude.

Next.
When balloon reached certain height then In first closed system using electrical energy from battery the compressor will start process of gas compression. It will use all electrical energy to compress gas to same balloon volume. The first system is closed and has efficiency 100%.
Plus extra potential energy of load from certain height.

No.
The balloon will NOT compress to same volume, unless extra energy is used to do so.
At this new certain height, the atmospheric pressure is exponentially less than it was at 'ground level'.
the balloon is pushing outwards against compression with a much greater force at this new altitude.
It is precisely equivalent to the gain in gravitational potential within the atmosphere.

[sm0ky2]

think of it like taking the square root of a square.

2^2 = 4
(sqrt)4 = 2

[abv]

think of it like taking the square root of a square.

2^2 = 4
(sqrt)4 = 2

Well, the pressure of atmospheric pressure changes by exponential function. However, the potential energy of load has a linear function. Not sure the differential of work against air on on altitude will increase linearly and it won't be equal to potential energy of load.

Looks like this is more complicated then 2x2.