# Confusion about forces in the work done on the piston by the gas in Carnot cycle

I was drawing a free body diagram of the 3rd step of the Carnot cycle as shown below.

The forces with - sign are reaction forces to the forces without the sign, according to Newton's 3rd law of motion. In this diagram, the forces applied on the piston from the gas are -Fpg and Fpres. Fpres is the pressure force due to collisions of gas particles on the piston, while -Fpg is originated from the piston which is pushing the gas; since the piston pushes the gas, the gas also pushes the piston in opposite direction. Since -Fpg and Fpres comes from different origins, so I think they're different force entities. So in the calculation of the work done on the piston by the gas, these forces are all to be included.

However, the thermodynamic textbook seems only taking Fpres into account so that the infinitesimal work is just PdV. Why is -Fpg not included in work calculation?

So, first, the work done on an ideal gas is always $PdV$. I really have no idea how your question relates to what is essentially a fundamental law of ideal gasses: you do work on them by changing the volume that they are constrained to. Nonetheless, I will try to explain what I perceive to be your misunderstanding.
I'm not super sure what $F_{pg}$ is referring to, but I think you are meaning the downward force of the piston (due to gravity or some external force) and will assume as such. I will also assume that $F_{pres}$ refers to the upward force of the gas caused by pressure.
• As I said, the gas does exert a force. It exerts a pressure force to counter the gravity force of the piston. $F_{pg} = - F_{press}$ Mar 21, 2017 at 1:48