A I'm trying to follow the proof given in Box 5.3 of MCP (Thorne/Blandford)

[jouvelot]

I'm trying to follow the proof given in Box 5.3, page 235, of the MCP book regarding the Van der Waals grand potential. It seems to me that there is a missing factor (2l−1)!/(l−1)! in the last term of Equation (8). What am I doing wrong?

 

[berkeman]

Can you post a high-quality scan image of that box? Use the "Attach files" link below the edit window to upload the image. Thanks.

 

[jouvelot]

Thanks @berkeman

Here is the page you requested (the attached version seems to be of rather poor quality). I don't think the previous page is needed, but if you think it could be useful, please let me know.
The original, good quality image is available at https://www.dropbox.com/scl/fi/ep2xqpo5ko9ybdancd33f/20240401_185542.jpg?rlkey=gnsw4ftd8ou2ea8konhgh6mn9&dl=0.

 

[TSny]

The circled $l!$ terms should each be $(2l)!$. The final line of (8), as given in the book, is correct.

 

[jouvelot]

View attachment 342684

The circled $l!$ terms should each be $(2l)!$. The final line of (8), as given in the book, is correct.

Ah, this makes a lot of sense, indeed. I was expecting Line 8 to be correct, since the rest of the proof follows through with this value, but couldn't find the error. Thanks a lot, @TSny, for spotting the error before that.