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I have not seen な used with a noun in this manner before (I know it can be
used with な "nouns" but I do not believe that 日曜日 is one of these) and was
wondering what is the exact effect of な in this passage and if there are any
particular grammar rules around its usage here.

今日は日曜日 **な** ので銀行は休みです: Banks are closed because today is Sunday.

Any help would be much appreciated.
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grammar particles
What is な’ｓfunction in this sentence?
「ので」 is used when we express resons (理由・原因). Connection (接続) of 「ので」 is 連体形 +
の.

> ［接助］《準体助詞「の」＋格助詞「で」から》

「ので」 comes from 準体助詞「の」 plus 格助詞「で」, which means 「の」 could be seen as a noun
here.

You might think 「日曜日 **の** ので」 is the right answer, but it's just too
difficult to say in daily converstion (because of the double の, it sounds
weird), so we use 「日曜日 **な** ので」 instead.
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### The grammar

Much as 古手梨花 described, ので is used to express a reason for something, similar
to "because" in English. Grammatically, ので is the conjunctive conjugation of
のだ (plain form) and のです (polite form), which can be parsed as possessive or
genitive の + the copula ("to be") verb だ or です.

の attaches directly to nouns or noun phrases, but because のだ describes
reasons, it attaches to statements. If you're making a statement about a noun,
you finish with the copula だ or です. And, grammatically, those statements that
のだ attaches to must be in the 連体形【れんたいけい】 or "attributive" form, and the
attributive form of だ or です is な.

Examples:

  * 本です。 (It's a book.)  
⇒ 本な[の]{●}[で]{●}[す]{●}。 (It's **because** it's a book.)  
⇒ 本な[の]{●}[で]{●}読みます。 (I read it **because** it's a book.)  
_(Note the reversed clause order to produce a more natural English
translation.)_

### Your sample sentence

> 今日は日曜日なので銀行は休みです。

Basically, this is a longer sentence with a shorter sentence embedded. Let's
break this down like the simpler example further above.

  * 今日【きょう】は日曜日【にちようび】です。 (Today is Sunday.)  
⇒ 今日【きょう】は日曜日【にちようび】な[の]{●}[で]{●}[す]{●}。 (It's **because** today is Sunday.)  
⇒ 今日【きょう】は日曜日【にちようび】な[の]{●}[で]{●}銀行【ぎんこう】は休【やす】みです。 (The bank is closed
**because** today is Sunday.)
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