How Much LN Can Shrink the Iron

Look up the linear thermal coefficient of expansion for cast iron. It is something like 10.8 X 10-6 m/m °C.

So find the difference in temperature in degrees C from its initial state to its final sate (immersed in the liquid nitrogen). Multiply that number of degrees by the coefficient of expansion and you are done.

I should add, the result in my first post must be multiplied by the dimension of the iron object.

< 14mm... IF your 14mm rod is not already at -200°C or so.

Is the volume on LN₂ sufficient to immerse this iron piece before it boils away?

Very roughly 13.7mm (assuming, among other things, that it measures 14mm at room temperature).

This sounds too much like homework to me.

Here is a site that will give you lots of information:

Online Materials Information Resource - MatWeb

This sounds like a marvelous technique for testing if the piece of iron has any mechanical imperfections and stress points. The thermal conductivity of iron is relatively poor for a metal, particularly at room temperature (300K). Immersing room temperature iron into LN2 at 40K will cause the outside of the iron to contract in size much faster than the center core of the iron.

I'm just a EE so correct me if I'm wrong, but isn't this a volume or area, as opposed to a linear coefficient of expansion problem? OP only gave us an area dimension and didn't mention the shape (wire, rod, washer, shim, sphere, etc.). According to Wikipedia the linear coefficient is 1/3, and the area coefficient is 2/3, that of the volume coefficient.

So which one do we use?

I think it is linear and in all directions. So, if you had a steel ball and lowered its temperature it would shrink in all dimensions uniformly based on its coefficient of expansion.

Since Iron has a body centered cubic crystal structure the volumetric expansion coefficient (β) will be three times the linear coefficient (α). Now depending on impurities and the shape this could just simply shrink as the OP wishes or internal stresses from rapid cooling could crack or shatter things as I proposed earlier.