A hole is drilled through the center of Earth and very long L=100km sealed cylindrical pipe?

Question

A hole is drilled through the center of Earth and very long L=100km sealed cylindrical pipe is inserted vertically into the hole. The pipe contains thermally insulated nitrogen gas, which is in equilibrium, has temperature T=400K, and pressure 1atm at the bottom.

Under these conditions the gas can be considered ideal, and pressure of the gas at the top of the pipe is negligible.
Its specific heat Cv = 5/2 R.

The pipe is initially at the surface, and then is very slowly (adiabatically) lowered downward the center of the earth. As gravity decreases, the gas is able to expand and reach higher levels inside the pipe.

What is temperature of the gas when the pipe is half-way down to the center of the earth?

norcekri:

I figured that accleration of gravity half way down will be g/2, using Gauss theorem.

I also figured that pressure at the bottom will be half of original pressure, because the weight of the gas in the pipe will be halved, and pressure at the top is negligible .

But here I stuck with equation Po/2 V = RT, becuse there are two unknowns and only one equation here.
Can you help me now?

Answer 1

The gas at the bottom expands and the gas at the top compresses, and there's no effect on the surroundings. This seems to be closest to a case of free expansion (exemplified by a chamber with a partition that can be opened). In such a case the temperature remains the same. (See ref.)

Answer 2

I would question whether a hole could be drilled through the center of the earth. The core believed to be composed mostly of iron, at a pressure of about 3.5 Mbar, temperature 7000K. How the hell do you drill in to that?

Answer 3

You haven't shown your work so far, so it's pretty hard to figure out where you're having trouble. Doing the entire problem for you would be cheating. Let us know how far you got?