Infinitive as a purpose or action happening at the same time that another.

(1) Here is the infinitives as a purpose:
I have come here to see her.
I left my village to enter university.
I went to the station to see off а friend.

(2) Here, the infinitives is used to show that the action expressed by the infinitive and the action of the verb happens simultaneously (unfortunately I don't know what name for this kind of adverbial phrase):
Hydrogen and oxygen unite to form water.
The atom may eject another particle to become a different atom.
Elements combine to produce a compound.
Attractions and repulsions often cooperate to reduce the viscosity of molecules.

Both the infinitives-(1) and infinitives-(2) are in the end of the sentence. I'm trying to find something that helps me distinguish one infinitive from another.
Couldn't you give me a little examples and I'll try to define what type of the infinitive it refer to?

Maybe I'm missing something but I don't see any real difference in these two situations. I could use the phrase "in order to" before all those sentences. I don't see the examples in (2) expressing simultaneity, rather processes.

I suppose you could say "in order to" in the first group, and "by which" in the second.

I have come here (in order) to see her. Some action takes place to achieve the result of being able to perform the second action. You cannot say "I have come here by which to see her".

Hydrogen and oxygen unite, by which they form water. The process of uniting itself has the result of achieving the transformation to water. It sounds weird if you say "in order to form water" because it's not exactly a plan: "Hey oxygen, let's unite and be water for a change".

Elements combine, by which (process) they become a compound.

Yes, you are right. I seems to have goofed. In (2), it looks like a result of previous action:

Hydrogen and oxygen unite, and water formed as a result of it.

At first I thought that infinitives (in the second type) is used as Present Participle in this sentence:

A girl sat reading a book.

That equals A girl was seating and reading a book. That is, these two actions happens simultaneously.

Thank you, bitpicker & sperk.

Quote:

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Originally Posted by oldboy

That equals A girl was seating and reading a book. That is, these two actions happens simultaneously.

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That would be "sitting". "Seating" is the act of showing someone to their seats, for instance in a theater.

Quote:

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Originally Posted by bitpicker

That would be "sitting". "Seating" is the act of showing someone to their seats, for instance in a theater.

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Yes, yes, I've made a mistake. Thanks.