Understaning articles:

The Curies, who had discovered ______ radium, were awarded ______ Nobel Prize for Physics.

The correct answer is "X" radium and "the" Nobel Prize.

My query: When we refer to Sun we say "The Sun" because there is only oneSun. Then why we don't use "The radium" since there is only one radium element?

They elected Lincoln ______ President of the United States.

Why not "The" President of United states? Why the article is omitted in this case?

In Europe, children begin to study ______ second language in primary school.

Why "a second language' ? What is indefinite in this case?

Thanks for your patience.

>The Curies, who had discovered radium, were awarded the Nobel Prize for Physics.

Radium is one of the many elements in the periodic table. In this sentence, there is no need to specify it to make it definite. Hence, we have not used an article.

While 'Sun' is a unique noun. 'The' is used before things unique of their kind.

>They elected Lincoln President of the United States.
This is a bit difficult to understand. Usually, 'the' is not used with words like 'elected'. We omit the article when we are mentioning the exact, precise title. Let us consider the given sentence:
They elected Lincoln, as what? (What: the exact title)
Answer: President of United State. 

>In Europe, children begin to study  a second language in primary school.

In this sentence, we are referring to 'second language' in general. It is not a specific, definite second language. Hence, we cannot say 'the' second language (there might not be only one second language).