Wir müssen heute einkaufen, denn der Kühlschrank ist leer.

Because the subject is wir (we), and the verb müssen has to agree with that subject.

Present tense of müssen:

• ich muss
• du musst
• er/sie/es muss
• wir müssen
• ihr müsst
• sie/Sie müssen

So Wir müssen = we must / we have to.

Because müssen is a modal verb. In German, when you use a modal verb with another verb, the modal verb is conjugated and goes in the normal verb position, while the main verb stays in the infinitive at the end of the clause.

So:

• Wir müssen heute einkaufen.

Structure:

• Wir = subject
• müssen = conjugated modal verb
• heute = time expression
• einkaufen = infinitive at the end

This is very common in German:

• Ich kann Deutsch sprechen.
• Sie will nach Hause gehen.

Yes. einkaufen is a separable verb: ein + kaufen.

That matters because:

• When it is used as the main verb, it separates:
  • Wir kaufen heute ein.
• When it is used with a modal verb, it stays together in the infinitive:
  • Wir müssen heute einkaufen.

So both are correct, but the form depends on the sentence structure.

In a normal main clause, German usually keeps the conjugated verb in second position. Other elements, like time words, often come after it.

So:

• Wir müssen heute einkaufen.

This is a very natural word order:

• subject
• conjugated verb
• time expression
• infinitive at the end

You could move heute for emphasis:

• Heute müssen wir einkaufen.

That also works, but then müssen still stays in second position.

Here, müssen means must or have to.

In everyday English, have to is often the most natural translation:

• Wir müssen heute einkaufen.
• We have to go shopping today.

It expresses necessity, not just desire.

Compare:

• Wir wollen heute einkaufen. = we want to shop today
• Wir müssen heute einkaufen. = we need to / have to shop today

Because denn connects two main clauses, and in German, clauses joined by denn are normally separated by a comma.

So:

• Wir müssen heute einkaufen, denn der Kühlschrank ist leer.

Each side could stand as its own sentence:

• Wir müssen heute einkaufen.
• Der Kühlschrank ist leer.

That is why the comma is used.

Denn means because in the sense of giving a reason.

Important difference:

• denn is a coordinating conjunction, so normal word order stays the same:

  • ..., denn der Kühlschrank ist leer.

• weil is a subordinating conjunction, so the verb goes to the end:

  • ..., weil der Kühlschrank leer ist.

So both can mean because, but the grammar is different.

Compare:

• Wir müssen heute einkaufen, denn der Kühlschrank ist leer.
• Wir müssen heute einkaufen, weil der Kühlschrank leer ist.

Both are correct.

Because der Kühlschrank is the subject of the second clause.

In:

• der Kühlschrank ist leer

the fridge is the thing that is empty, so it must be in the nominative case.

For the masculine noun Kühlschrank:

• nominative: der Kühlschrank
• accusative: den Kühlschrank

You would use den Kühlschrank only if it were a direct object, for example:

• Ich putze den Kühlschrank.

Because leer is a predicate adjective here, not an adjective directly before a noun.

After verbs like sein, werden, and bleiben, adjectives usually do not take endings:

• Der Kühlschrank ist leer.
• Das Wasser ist kalt.
• Die Wohnung ist teuer.

But before a noun, adjective endings do appear:

• der leere Kühlschrank
• das kalte Wasser
• die teure Wohnung

So:

• Der Kühlschrank ist leer. is correct.

Because all nouns are capitalized in German.

So:

• wir is not capitalized unless it starts the sentence
• müssen is not capitalized
• heute is not capitalized
• Kühlschrank is capitalized because it is a noun

This is a basic and very important rule in German spelling.

Yes, it is one word. German often combines nouns into compound nouns.

Kühlschrank is made up of:

• kühlen / Kühl- = cooling
• Schrank = cupboard / cabinet

So literally it is something like a cooling cabinet, which is why it means fridge / refrigerator.

German compounds are very common, and they are usually written as one word.

You probably could in some contexts, but it would not mean exactly the same thing.

einkaufen usually means to shop, especially for everyday necessities like food or groceries.

Because the sentence mentions der Kühlschrank ist leer, einkaufen is the natural choice.

shoppen exists in German too, but it often sounds more like shopping for fun, clothes, or leisure.

So in this sentence, einkaufen is definitely better.

Yes:

• Heute müssen wir einkaufen, denn der Kühlschrank ist leer.

That is completely correct. German allows different elements to come first for emphasis, but the conjugated verb still stays in second position.

So:

• Wir müssen heute einkaufen ...
• Heute müssen wir einkaufen ...

Both are natural. The second version emphasizes today a bit more.

Because the clause after denn is still a normal main clause, and in a German main clause the conjugated verb is in second position.

Second clause:

• der Kühlschrank = first element
• ist = conjugated verb in second position
• leer = adjective

So:

• der Kühlschrank ist leer

If you used weil instead, the verb would move to the end:

• ..., weil der Kühlschrank leer ist.

Not exactly.

• Wir kaufen heute ein. = We are shopping today / We shop today.
• Wir müssen heute einkaufen. = We have to shop today.

The second sentence adds the idea of necessity or obligation.

So the action is similar, but the meaning is not identical.