Gas - cylinder - piston problem

[akmkeng]

Summary:: Seeking explanation to classical gas - cylinder - piston problem, not the solution.
Problem 1.15 from 7th edition of Introduction to Chemical Engineering Thermodynamics by Smith, Van nes and Abbot)

Classical problem, given:
- gas in a confined cylinder
- piston with weight is placed above
- no friction between piston and cylinder
- weight of the piston+weight
- atmospheric pressure
(This 1.15 from 7th edition of Introduction to Chemical Engineering Thermodynamics by Smith, Van nes and Abbot)

(a) asks for total force acting on gas by Atmosphere, piston+weight.
F_total = P_atmA_piston + m_{piston+weight}g

(b) This is straghtforward force balance:
P_gasA_piston = P_atmA_piston + m_{piston+weight}g

(c) Gas is heated and expands. Displacement of piston+weight is given. Need to find Work and ΔE_p.
Now I understand the dW = Fdl, and by explanation I found elsewhere on internet, the F should be at least equal to total Force that is being acted on gas by atm, piston+weight (from point a), and this makes sense to me. ΔE_p is straightforward from ΔE_p = mgdl as well.

What I don't understand is say, we heated the gas, it expanded and the system is kept in this state. Let's say the gas properties before this are state 1 and after - state 2. After that, let it be the moment we started solving this problem. Repeating point (b) we will find the P_gas, but it will be equal to P₁, whereas we know it was heated and expanded and is now P₂. This is where my brain trips and everything else starts not making sense.

Please help out. I am redoing undergraduate level problems to gain stronger basics understanding and don't seem to be able to skip this one. Please.

I attached the schematics as I see it, sorry for poor quality, if so. Also I see I forgot to indicate P_atm in my scheme, but it doesn't change anything much, so leaving it as it is.

 

[akmkeng]

I just had the realization, is the system with changed gas pressure balanced by changed atmospheric pressure?
so before heating it is:
P_gas1 A_piston = P_atm1A_piston + m_{piston+weight}g
after heating:
P_gas2A_piston = P_atm2A_piston + m_{piston+weight}g

But it doesn't correlate with understanding that the atmosphere is surroundings and too big to have been impacted by piston movement. However, is it not? Considering it is given in the question as one of the constituents of system as well. I see that my thinking contradicts itself here, atmoshhere is both system and surroundings according to me. Let's establish it as part of the system, then the question is it constant or not? In which cases can we consider it possible to be affected by system and when it is not?