Barometer Tube Displacement: Solving for Isothermal Process

[Tidus]

Homework Statement

A thin tube, sealed at both ends, is 1.00 m long. It initially lies horizontally, with the middle 10.0 cm containing mercury, and the two ends containing air at standard atmospheric pressure:

See picture in attachment

If the tube is now turned to a vertical position, by how much will the mercury be displaced, if the process is isothermal? You may assume that no gas passes from one side of the mercury column to the other.

Homework Equations

P_1f = hpg + P_2f p=density of mercury , h=length of mercury column
P₁V₁ = P₂V₂
V₁ + V₂ = 0.9

The Attempt at a Solution

Tried solving simultaneous equations but got bogus answers

 

[Doc Al]

Say the mercury is displaced X cm. How much does the pressure change on each side? How much of a pressure difference is needed to support the mercury?

 

[Tidus]

(P₂ + hpg)(0.45-x) = (0.45 +x)P₂ ?

 

[Doc Al]

(P₂ + hpg)(0.45-x) = (0.45 +x)P₂ ?

Not sure how you got that expression. Do it step by step. Both sides start with the same volume (proportional to their length, which is initially 0.45m) and the same pressure Pa.

So what's the new pressure of the air on the side that compresses by distance X? And the side that expands?