My apologies for not keeping track of this post sooner. I appreciate all of the responses and effort put into this question!
I will try and explain my approach to this problem. Studiot, I reviewed my energy balance and agree that the kinetic energy term in my original equation is not needed here. My energy balance has become dPE = dWs where, PE and Ws are explained in my original post.
dWs = PLdVL - PRdVR = nRTL/VLdVL - nRTR/VRdVR (2)
where PL and VL are Pressure and Volume of the Left hand chamber while PR and VR are Pressure and Volume of the Right hand chamber.
Since this is an isothermal process, PLVL = constant at any point and is actually 800,000 N/m^2*0.1m^3 = 80,000 N*m (or Joules). Same can be said of PRVR = constant = 100,000 N/m^2*0.15m^3 = 15,000N*m (J).
From the Ideal Gas Law nRTL = 80,000 J while nRTR = 15,000 J (Since this is an isothemal process)
Then, dWs = 80,000/VL*dVL – 15,000/ VR*dVR
And, PE = Ws = 80,000*ln(VL2/VL1) - 15,000*ln(VR2/VR1)
But, since we already know VL1 = 0.1 m^3 and VR1 = 0.15 m^3 (Assuming the thickness of the piston has been taken into account when the question said the total volume of the cylinder is 0.25 m^3…I am assuming that is the total open space volume)
PE = Ws = 80,000*ln(VL2/0.1) - 15,000*ln(VR2/0.15)
And since we know that VL+VR = 0.25 m^3, then VR2 = 0.25 -VL2 and the equation becomes
PE = Ws = 80,000*ln(VL2/0.1) - 15,000*ln((0.25-VL2)/0.15)…
Now, I have 2 unknowns PE, and VL2.
This is where I am stuck…..if I can get the Potential energy associated with this then I can calculate VL2 which will allow me to calculate the position of the piston and Pressure (from the ideal gas law)……This is what I have done so far…please feel free to show me any mistakes in my thinking or, on the other hand what I might do to calculate the PE…seems I need to know the final position of the piston before I do that…..I will try to respond in a more timely manner this time around!
Thanks!