Does iron increase in weight when it rusts?

The answer is yes and no. Let’s consider the yes part first. What happens when iron rusts is that the outside of it, which is exposed to the air, corrodes or oxidizes. A certain amount of oxygen of the air is added to the iron. This oxygen, like everything else, has weight, and its weight (actually relative atomic mass of 15.9994) must be added to that of the iron itself when the iron is rusty. Therefore, the answer to the question must be yes. Iron increases in weight. But as everyone knows, the rust, or oxide of iron, is friable; a Latin word which means that it crumbles easily.

Now the answer which is clearly no. The rust will crumble away under the influence of water or wind or anything else rubbing against the iron. Therefore, the iron object will lose not only the oxygen that it has taken into itself, but also the part of the iron which has combined with oxygen. So the material made of iron, when it rusts, loses weight. The whole piece crumbles into red dust eventually. This is very serious, of course, for it means that the object loses its strength. And if an iron and steel bridge were allowed to decay in this fashion, it would soon break. That is one reason why such a bridge is painted to protect the iron from the air.

Additional reading:
Rust (Wikipedia)
Iron (Wikipedia)

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