# Parallel lines at infinity

I have to find the maximum work done in expanding gas to twice its volume.

I imagined a P vs V graph and took a point A at volume V and I have to do some process that would take it to point B at twice its volume i.e. 2 V

Work done is equal to the area under the line AB.

For maximum work the line AB would cover largest area.

I visualised the line AB rotating about point A and its area is increasing when I am rotating it. At 0° it is rectangle, at 45° it is trapezium, at 89.99° it is 'very big' trapezium and at exact 90° it should be logically even bigger. That's what my mathematical knowledge say that parallel line meet at infinity.

Then I thought of any process that would make 90° angle and that is isochoric process.

But then my brain said that how could be there work done in isochoric process as volume is constant and it is necessary for volume to change for any work done. Also how could we increase volume by a process that is famous for constant volume.

Please have a look at my dilemma.

And sorry for my exceptional drawing skill :( .

• This is a math question, not a physics question. Commented Oct 31, 2023 at 10:31