Die Glühbirne im Flur ist kaputt, deshalb kauft meine Schwester eine neue.

Because Glühbirne is a feminine noun in German, so its basic article is die.

• die Glühbirne = the light bulb
• singular feminine nouns often use:
  • die in the nominative
  • die again in the accusative

In this sentence, die Glühbirne is the subject of ist kaputt, so it is in the nominative case.

Glühbirne means light bulb.

Literally, it is made up of:

• glüh- = glowing / incandescent
• Birne = pear

So the original image is something like a glowing pear-shaped bulb.

In everyday German, Glühbirne is a very normal word. Even though many modern bulbs are not technically old-style incandescent bulbs, people still commonly use Glühbirne to mean light bulb in general.

Im is a contraction of in dem:

• in dem Flur → im Flur

German uses in + dative here because the sentence describes a location:

• Die Glühbirne ist im Flur. = The light bulb is in the hallway.

This answers where?, not where to?

So:

• im Flur = in the hallway / in the corridor
• case: dative

Because German distinguishes between:

• location → dative
• movement toward a destination → accusative

Here, the bulb is located in the hallway, so German uses dative:

• im Flur = in the hallway

Compare:

• Die Glühbirne ist im Flur. = The bulb is in the hallway.
• Ich gehe in den Flur. = I go into the hallway.

So im Flur is correct because there is no movement.

Because kaputt is being used as a predicate adjective after sein.

In German, adjectives used this way do not take adjective endings:

• Die Glühbirne ist kaputt.
• Der Flur ist dunkel.
• Meine Schwester ist müde.

Adjective endings are only added when the adjective comes before a noun:

• eine kaputte Glühbirne = a broken light bulb
• eine neue Glühbirne = a new light bulb

So:

• ist kaputt = correct
• ist kaputte = incorrect

Because this sentence contains two main clauses:

1. Die Glühbirne im Flur ist kaputt
2. deshalb kauft meine Schwester eine neue

In German, when two independent main clauses are linked like this, a comma is normally used.

So the comma helps separate:

• the reason/problem: the bulb is broken
• the result/consequence: the sister buys a new one

Because German main clauses follow the verb-second (V2) rule.

That means the finite verb must be in the second position. If deshalb comes first, the verb must come immediately after it:

• Deshalb kauft meine Schwester eine neue.

Structure:

1. Deshalb
2. kauft
3. meine Schwester
4. eine neue

This is very different from English.

Compare:

• Meine Schwester kauft eine neue.
• Deshalb kauft meine Schwester eine neue.

When deshalb moves into the first position, the verb still stays in second position, so the subject moves after the verb.

Because meine Schwester is the subject of the second clause, so it is in the nominative case.

The subject is the person doing the action:

• meine Schwester kauft = my sister buys

For a feminine noun phrase with mein- in the nominative singular, you use:

• meine Schwester

Compare a few forms:

• meine Schwester = nominative / accusative
• meiner Schwester = dative / genitive

So here:

• meine Schwester is correct because she is the one buying the new bulb.

Because German, like English, often omits a noun when it is obvious from context.

Here, eine neue clearly means:

• eine neue Glühbirne = a new light bulb

Since Glühbirne was already mentioned, repeating it is unnecessary.

The adjective still keeps the ending it would have before the noun:

• eine neue Glühbirne
• shortened: eine neue

This is very natural German.

Because the omitted noun is Glühbirne, which is feminine.

The phrase is in the accusative case because it is the object of kauft, but for a feminine singular noun, the article and adjective look like this:

• nominative: eine neue Glühbirne
• accusative: eine neue Glühbirne

So feminine singular stays eine neue in both nominative and accusative.

If the noun were masculine, it would change:

• ein neuer Tisch = nominative
• einen neuen Tisch = accusative

But Glühbirne is feminine, so eine neue is correct.

Deshalb means therefore, that’s why, or for that reason.

In this sentence, it introduces a result:

• The bulb is broken.
• Therefore, my sister buys a new one.

You can often replace it with similar words such as:

• deswegen
• darum

For example:

• Die Glühbirne im Flur ist kaputt, deshalb kauft meine Schwester eine neue.
• ..., deswegen kauft meine Schwester eine neue.
• ..., darum kauft meine Schwester eine neue.

All are natural, though deshalb and deswegen are especially common.

Yes, but the structure would change.

With deshalb, you have two main clauses:

• Die Glühbirne im Flur ist kaputt, deshalb kauft meine Schwester eine neue.

With weil, the cause goes into a subordinate clause, and the verb moves to the end of that clause:

• Meine Schwester kauft eine neue, weil die Glühbirne im Flur kaputt ist.

Notice the difference:

• deshalb → main clause word order, verb second
• weil → subordinate clause word order, verb at the end

Both are correct, but the grammar is different.