のに

1. Adversative usage, with an implied tone of surprise or dissatisfaction. English approximation: although, but, even though.

あら、もう三杯も食べたのに、またおかわり。My goodness, you've already had three helpings, but you want another one?
私の帽子、確かにここに置いたのに、ないわ。I know I definitely put my hat here, but it's gone.
夏休みだというのに、宿題がたくさんあって遊びに行けない。Even though it's (called) summer vacation, (I) can't go out to pay because (I) have too much homework.

2. Expresses dissatisfaction or recentment toward an unexpected situation. English approximation: (if...), can/would do ..., (but...).

君が来るとわかっていれば、僕、家にいたのに。If I had known that you were coming over, I would have been home, (but) ...
もう百円あれば、この本が買えるのになあ。If only I had one hundred more yen, I could buy this book, (but) ...

3. In rebuking the party addressed for a fault or wrongdoing. English approximation: eve though.

食べないの。せっかく作ったのに。You're not going to eat it, even though I went to the trouble of making it?

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のーに
Combination particle

1. Indicates the method by which an end is to be achieved. English approximation: in order to..., in doing...

日本語がもっと上手に話せるようになるのには、どうしったらいいですか。In order to become a better speaker of Japanese, what should I do?
山歩きをするのには、虫よけを持って行ったほうがいいよ。In order to hike mountains, it would be better to bring an insect repellent.

2. Indicates the means or material with which something is to be achieved. English approximation: for (i.e. as a means for doing...)

この箱、本を入れるのにちょうどいいね。This box is perfect for putting books in it, isn't it?
この貯金、旅行するのに少し足りないわね。This savings aren't quite enough for going on a trip, (don't you think)?

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Источник: A dictionary of japanese particles. Sue A. Kawashima. Kodansha international.