We have a Carnot cycle and lets say it is in thermal and mechanical equilibrium with the surroundings before the beginning of the process. I am using the convention that heat brought to the system is positive and the work done by the system is positive.
After the isothermal expansion has happened, isentropic expansion starts. The ideal gas does some more work, in turn cooling itself. The question is why this happens? What "forces" the gas to to some more work during the isentropic expansion?
When looking at this answer to the similar question, the answer seems to be that, it is true that the mechanical equilibrium was present at the beginning, but now we gradually decrease the pressure from the outside and thus inducing a pressure gradient for the gas to be able to expand, use its kinetic/internal energy to push the piston outwards, creating work and thus cooling itself.
And the answerer says: "Imagine a vertically oriented cylinder fitted with a piston on top of which as a bag of sand that provides the external pressure. Now imagine removing one grain of sand at a time which reduces the external pressure infinitesimally allowing the gas to expand infinitesimally. Continue the process until the final desired external pressure is reached."
Again the same logic is used in the answer about the work done during the isothermal expansion.
My question: We have to put in some work to move the infinitesimal grain of sand from the disk (to make it lighter) and thus letting the gas expand an infinitesimal amount and in turn producing some infinitesimal work. Now, for this process to make sense, we would have net positive $W$, how does this happen if we exchange the work put in for moving the grain of sand for the work done by the gas? Wouldn't the total work change during the isentropic expansion be zero than?
Could someone provide the equation for the work done by the gas during isentropic expansion.
Appendix on the Work-energy principle:
When I have a ball that is stationary and I use positive force on it and create positive work on it. It starts moving and lets say there is no friction. Than I go on the other side of the ball and I slow it down, i.e. I stop it and put it to a standstill. Than $\Delta E_{k}$ is zero so no work is done:
$$W=\Delta E_{k}=0$$
But, for me, as a human, I did work when accelerating it and when I was breaking it. How come is $\Delta W =0$ ? What does it mean that I did negative work when it sure didn't make me feel more energized, it just made me do more work. I was twice as tired, its not as I was resting and being charged up in the second case, where I was stopping the ball.
I understand that the work is by definition zero, because:
$${\Delta W=\int _{C}\mathbf {F} \cdot d\mathbf {s} +\int _{C}\mathbf {-F} \cdot d\mathbf {s} =0.}$$
But the question remains: How did the work I did during the continuous removal of the grains (to let the isentropic expansion take place) of sand end up being zero for me when I was getting tired during accelerating and decelerating the grain of sand?
Can I explain it in the way: Lets say I have an electric motor that does positive work on the particle, than the particle continues moving and the motor is put on the particles path and the particle powers the motor and puts the kinetic energy, i.e. the work that was done by the motor to the particle back into the motor by reabsorbing its moving energy (kind of motor breaking). The particle was used as sort of a just temporary holder of kinetic energy, just for the motor to absorb it back again and thus $W_{electric motor}=0.$
How did Carnot explain the fact that $W= 0$ if he didn't have this sort of argument I used with the electric motor reabsorbing its energy, i.e. "absorbing" back work?
