This is a doubt on question 5.3 from the 5th chapter of the book Heat and Thermodynamics by Zemansky and Dittman. The problem statement is as follows:
Mercury is poured into the open end of a j-shaped tube, closed at the short end, trapping air in that end. How much mercury can be poured in before it overflows? The length of the tube is given as 1.5 meters, effects due to curvature of the bottom can be neglected, and 1 atm = 76 cm Hg.
I get that at some point after pouring in mercury, the pressure of air trapped inside is so much that you can't further add mercury, thus leading to spilling. I also tried equating the product of the initial pressure and volume to the final pressure and volume of the air column. I took $P_i=76 cm Hg$ and $V_i=(150cm)(Area)$ and equated it to the final product(assuming isothermal conditions hold), taking the length of the air column as x cm, and final pressure as the sum of atmospheric and the pressure due to mercury column corresponding to the same horizontal level as the air column inside. But, the answer I am getting is 60 cm, which cannot be true as the shorter arm is given to be only 50 cm in length, and the air is trapped in the short end.
Could someone explain where am I wrong?
Edit: I am attaching a diagram as well. Here, x is the length of the air column and the shaded region is mercury filled in the tube. Initially, this tube was empty, open to atmosphere.