nouns followed by infinitives

Question

I have sometimes problem understanding these structures. I just came across this expression:

Materials to make sarin gas and perhaps other toxic chemicals moved freely cross-border from Turkey to Syria.

"Materials to make sarin gas and perhaps other toxic chemicals ." Here we can sense that materials are not the subject which will make sarin and other chemicals by itself.In this context , we can infer that the infinitive clause is used in a way that the meaning is close to "materials which is necessary to make sarin gas and other chemicals".

I will give another example and hence my confusion will be revealed.

The woman to save you
The institution to supervise the process

In these examples , we can safely infer that the subject is nouns which are both followed by infinitive clause. The woman is subject. The institution is subject.

So why there is difference? Am i missing a point? In the very first example I gave , the subject is not clear as it is understood from the context.We don't know the subject. Genereally , in the pattern of noun+to+verb+object i want to be enlightened as to subject-object relations of this pattern.

Answer 1

In your original sentence, "Materials to make sarin gas and perhaps other toxic chemicals moved freely cross-border from Turkey to Syria", yes, "materials" is the subject of the sentence. If you look carefully at the sentence, you will see that at its heart it is a simple subject-verb sentence: Materials moved. Everything else in the sentence is explanatory:
Materials
what kind? to make sarin gas and perhaps other toxic chemicals
moved
how did they move? freely cross-border from Turkey to Syria.

Your understanding of the bolded part of the sentence is basically correct, though. The construction noun verb-infinitive, like materials to make sarin gas or the woman to save you or the organization to supervise the process means essentially noun for the purpose of verb-ing. The materials for the purpose of making sarin gas moved freely; the woman for the purpose of saving you did whatever; the organization for the purpose of supervising the process did whatever.

Answer 2

Ingredients to make a birthday cake, and what looked like party hats, were in the shopping basket.

It's not clear whether all of the ingredients required to make a cake were in the basket (maybe sugar was missing, for example, or baking powder), but the ingredients in the basket could be described as those that go into making a cake.

The clause "and what looked like party hats" is simply additional information, unrelated to the ingredients.