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I read that zirconium is not very reactive even when supplied with a current in an electrolyte. If I had a sizable piece of it, can I use it to perform electrolysis for rust removal of items? Would it have the same or similar results to graphite which results in a less messy process, namely the black film left on parts? Is it toxic?

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No, you cannot use zirconium as an anode for electrolytic rust removal.

Zirconium is a member of the family of 'valve metals' (an archaic term), which form a strongly passivating oxide film when exposed to air and/or water. From here:

Owing to their low electrochemical potential the group IVB and VB valve metals Ti, Zr, Hf, V, Nb and Ta readily react with water or oxygen to form a dense, protecting passive layer.

You're right that these metals, including Zr, tend to be exceptionally chemically resistant (same source):

Because of these protecting oxide films, the valve metals show an exceptional resistance towards corrosion in many aggressive environments. This explains why valve metals are widely used in the construction of chemical apparatus.

However, a feature of these metals is that the resistance of the passivating oxide film increases significantly when an anodic bias is applied to them (same source):

Valve metal oxides are formed conventionally electrochemically ... .

The Wikipedia article on anodizing concurs (emphasis added):

Anodic films are most commonly applied to protect aluminium alloys, although processes also exist for titanium, zinc, magnesium, niobium, zirconium, hafnium, and tantalum.

Thus, were you to construct an electrochemical cell with a zirconium anode and a rusty steel cathode, I expect you would get a brief current transient as the Zr-oxide passive film grew a bit, but then it would tail off to essentially zero. While I haven't worked specifically with zirconium, the above is the observed behavior of all of the valve metals I have experience with: tantalum, niobium, titanium, and aluminum. I would be quite surprised if zirconium behaved any differently.

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