В библиотеке студенты шепчут, чтобы не мешать другим.

Because в changes case depending on meaning:

• в библиотеке = in / at the library → location, so Russian uses the prepositional case
• в библиотеку = into / to the library → direction or movement, so Russian uses the accusative case

The base form is библиотека, and in the prepositional it becomes библиотеке.

Russian word order is much more flexible than English word order.

Starting with В библиотеке sets the scene first: In the library...
It feels natural because it tells you the setting before the action.

A more neutral version like Студенты шепчут в библиотеке... is also possible, but the original order emphasizes the place.

Russian has no articles, so there is no separate word for the or a/an.

So:

• в библиотеке can mean in a library or in the library
• студенты can mean students or the students

The exact meaning comes from context. In this sentence, English naturally uses the library.

Студенты is the nominative plural of студент.

That means:

• студент = student
• студенты = students

It is in the nominative because it is the subject of the sentence — the people doing the action.

Шепчут is the 3rd person plural present tense form of шептать, meaning to whisper.

So it means:

• they whisper
• or, depending on context, they are whispering

Russian present tense often covers both the simple present and the present continuous in English.

Also, the stem changes a little:

• шептать
• шепчут

This kind of consonant change is normal in Russian verb conjugation.

Because студенты already tells you who is doing the action.

Russian usually does not add a pronoun if the subject noun is already there:

• Студенты шепчут = The students are whispering

Adding они would usually sound unnecessary unless you want emphasis or contrast.

Here чтобы introduces a purpose clause.

It means something like:

• so that
• in order to

So чтобы не мешать другим explains why the students whisper.

Because the subject is the same in both parts of the sentence.

• студенты шепчут
• the same students are the ones who do не мешать

When the subject stays the same, Russian often uses чтобы + infinitive:

• чтобы не мешать другим = so as not to disturb others

If the subject were different, Russian would usually use a finite verb instead:

• Студенты шепчут, чтобы другие не слышали.
• The students whisper so that others won’t hear.

Мешать is the imperfective verb, and it fits well here because the idea is general: the students whisper so that they do not disturb others.

This is not about one single completed act of disturbing someone. It is about avoiding disturbance in general, so мешать is the natural choice.

Very roughly:

• мешать = to disturb / bother / interfere with
• помешать = to hinder / prevent / disturb in a more single-result sense

Because the verb мешать takes the dative.

You can think of it as:

• мешать кому? = to bother/disturb whom?

So:

• другим = to others

Grammatically, другим is the dative plural form of другой.

In Russian, adjectives can sometimes be used by themselves like nouns. This is happening here.

So другим literally means to other [people], but the noun is omitted because it is understood from context.

You could expand it to:

• другим людям = to other people

But другим by itself is completely natural.

Because чтобы не мешать другим is a subordinate clause, and Russian normally separates such clauses with a comma.

So the comma is required:

• В библиотеке студенты шепчут, чтобы не мешать другим.

This is standard Russian punctuation with чтобы when it introduces purpose.