The entropy of the surroundings does change infinitesimally. But the surroundings are large and such a change does not change the total entropy of the surroundings in any sensible way.
Indeed, one already uses that fact in putting the system through a series of reversible steps. As you point out, if the temperature of system and surrounding were in fact identical, no heat would flow. But they are infinitesimally different and so an infinitesimal amount of heat does flow.
The same applies to the surroundings. It too is undergoing a reversible change.
1) If we take the heat out of the system (i.e. with a cooling circuit, like happens in car engines) would gas temperature go down to the initial state, without decompressing it?
Is the piston fixed? Then the volume will remain constant during the cooling. Gas temperature would go down and pressure will fall.
How far down the temperature will go depends on the cooling. If the cooling circuit is bound to a lower temperature, then it will continue cooling until the gas has reached that. Independent of any initial temperature of the gas.
2) How long could you repeat that compression-heating-cooling process? Would it result in liquefying the gas?
If I understand it correctly, you remove the heat caused by compression (and return to initial temperature), then compress it again, then remove the heat and repeats like that.
This is exactly what happens in a usual pressure cylinder - like and air cylinder for diving. That will have a large pressure inside because a lot of air is compressed, but the temperature is equal to the surroundings.
Yes, it will liquify at some point. You compress it, pressure rises and the boiling point also rises. It will stay as gas, if the temperature is allowed to rise also. But by removing heat and cooling it down, you will at some point have a temperature below the new boiling point. In a phase diagram of a substance you will be able to see where the transition points are from liquid to gas - that point is tightly bound to volume, pressure and temperature.
3) How can I calculate the relation between the piston speed, pressure, and the heat generated?
Why piston speed? If you move the piston fast, you add kinetic energy to the molecules not caused by the volume decrease. You will have a hard time figuring out how much is due to piston movement.
Usually in models of this kind of volume decrease in a pressure champer the piston is assumed to move so slowly that kinetic energy is negligible. Then piston speed will have no relation to the other parameters.
Best Answer
No. To increase the speed of the car, you do are 2 things: you press the accelerator, and you change gears.
Pressing the accelerator makes the engine turn faster, even when the car is out of gear and stationary. You can see the engine speed on the tachometer, which shows the Revolutions per Minute (RPM) of the engine. This is the number of times the crankshaft rotates per second. Typical figures are between 500 rpm and 7000 rpm. As every cylinder has to go up and down once for each revolution, they obviously move faster as you press the accelerator pedal further down.
You also need gears because an internal combustion engine has very little torque and power at low rpm, while it also cannot turn faster than some maximum speed without self-destructing. Hence, you use low gear at low speed (to have the engine run fast enough so it has enough power at low speed) and you use high gear at high speed (so the engine won't have to turn at excessive rpm). A good driver will choose the correct gear, depending on what they want to do. To accelerate fast, use a lower gear; to use less petrol, use a higher gear.
If the tacho in your car does not change much, that is because you have an automatic gearbox (or even a CVT gearbox), which changes gears to keep the engine running at optimum speed. If you manually select a gear (or if you drive really fast) you'll see the tacho change.
Note that electric cars do not have this problem. Electric motors have maximum torque at zero rpm. Hence, all you need to do is to set the overall gearing so that the car will go at maximum speed at the recommended maximum rpm of the motor. You do not need to worry about having the motor run at speeds that are too low: it can "run" at 0 rpm; this is unlike a petrol engine which would stall.