[katieHaylor (OP)]

Jim asks:

If one could drill a hole through the center of the earth, what would be the air pressure at the center where gravity is zero? And how would the pressure vary with depth?

Any thoughts?

[Halc]

If one could drill a hole through the center of the earth, what would be the air pressure at the center where gravity is zero? And how would the pressure vary with depth?

Assuming the hole is small enough not to reduce the volume of the atmosphere significantly, one would have to integrate the increasing density of air as the pressure grew with depth over the decrease in gravity (and thus the gradient of the pressure differential) as the center of Earth is approached.

So the answer is "a lot".  Pressure at the surface is around 100 kPa, and even 5 km down it would be twice that already, so I figure the pressure at the center to be 3-4 orders of magnitude greater than it is at the surface.

The lack of gravitational gradient at the center would mean that the pressure is fairly uniform there, not changing significantly at first as one moves from the center to the surface.

[evan_au]

a lot

Would the pressure be high enough to turn air into a liquid (or solid)?

[Halc]

Would the pressure be high enough to turn air into a liquid (or solid)?

The temperatures at those depths would exceed the critical temperature before that happens, so you'd end up with a supercritical fluid once pressure also crosses that barrier. Read up on the conditions on Saturn, which does kind of exactly this, and of all the planets, has surface gravity closest to that of Earth. Of course the atmosphere mixture is very different than ours.
There seems to be no hope of a solid, which at those pressures might be present at temperatures a 20th of that found deep within Earth. It exceeds the temperature of the surface of the sun there. Makes you wonder what we're going to line our hole with to allow the air to get down there. What material can take that sort of abuse?

I'm not entirely familiar with the compressibility properties of supercritical substances, but it seems that at the temperatures likely that deep down, the density curve is fairly linear as gas goes supercritical and retains most of its gas-like properties. At lower temperatures where it is more liquid-like, the density goes up rapidly at the critical point but then falls off.

Any integration of the pressure curve would have to take supercritical properties, temperature curve, as well as the real gravity-gradient into account. To be fancy, you'd also have to model the more dense gasses that are likely to collect in the hole, buoying up the other elements.  The air down there would have a very different mixture than here at the surface.

[Bored chemist]

What material can take that sort of abuse?

Unobtanium.
As you know, it's available in two varieties, the thermal insulating version, and the thermally conductive version.
It's probably easier to do the maths with the insulating version.
The pressure due to rocks above the centre of the Earth is about  3 million atmospheres.
There are two approximations you can make.
The first idea is that, since air is about 1,000 times less dense, the pressure will be about 1,000 times less.
But air compressed to 3000 atmospheres will have a much higher density- if it followed the ideal gas laws it would be 3000 times denser than at normal pressures.

Which means it's nearly as dense as the rocks.
Which means that, to a second approximation, the pressure in the middle will be pretty similar to that due to rocks pressing down.

In other words, it doesn't matter very much what you fill the hole with.
The pressure is going to be more or less the same.

[alancalverd]

Er,

http://hyperphysics.phy-astr.gsu.edu/hbase/Mechanics/sphshell2.html

suggests that the air deep inside the hole is weightless.

[Bored chemist]

Er,

http://hyperphysics.phy-astr.gsu.edu/hbase/Mechanics/sphshell2.html

suggests that the air deep inside the hole is weightless.

The clever bit is that the same applies to the rocks and they add up to about 3 MBar
That's why I picked this datum.
Most of the maths- the calculus bits- have been done for me.

[evan_au]

The 2012 version of Total Recall imagined a tunnel bored through the Earth from UK to Australia, and commuters traveling through it to get to work.

Ignoring the need for a considerable supply of thermally-insulating unobtanium (which, according to Avatar is available near Alpha Centauri), the tunnel would need to be kept under vacuum, or the commuters would quickly come to a relatively small terminal velocity, which would not get them to the other end of the tunnel.

There were also a few other quirks in the movie: the commuters seem to experience normal gravity for most of the journey, and are only in free fall for a few seconds at the center of the Earth.
...but what use is science fiction if it doesn't get you thinking?

Spoiler Alert: https://entertainment.time.com/2012/08/06/spoiler-alert-the-8000-mile-hole-in-total-recall/

[Halc]

Er,

http://hyperphysics.phy-astr.gsu.edu/hbase/Mechanics/sphshell2.html

suggests that the air deep inside the hole is weightless.

Of course it is, but the OP asks about pressure, not the weight of the small portion of air that's at the center, and the pressure is due to the weight of the entire column, not just the bit at the center.

[alancalverd]

Inside a homogeneous sphere, g = 0 everywhere. Weight = mg. So to a first-order approximation  it's only the air above the surface that determines the pressure inside. 

The reason I said "deep inside..." is because the core is denser than the crust but once you are inside the mantle it's pretty homogeneous.

[Bored chemist]

Inside a homogeneous sphere, g = 0 everywhere. Weight = mg. So to a first-order approximation  it's only the air above the surface that determines the pressure inside. 

The reason I said "deep inside..." is because the core is denser than the crust but once you are inside the mantle it's pretty homogeneous.

By that argument, the pressure at the bottom of the ocean is the same as at sea level.
Meanwhile, for those of us who are not "hollow Earthers"...
You have used the wrong formula (gosh) and come up with an impossible outcome, but not noticed.
The Earth is not a spherical shell.
There's some explanation here of why the acceleration due to gravity rises (or falls) linearly with radius.
https://cnx.org/contents/1AJmkLE0@2.9:XnS_IoKu@2/Gravitational-field-due-to-rigid-bodies

Essentially, imagine peeling a spherical layer off the Earth.
As you have said, that layer contributes nothing to the gravity inside it.
But the rock inside the shell still behaves like a (smaller) planet and it has a net pull downwards.