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Temperature and vacuum

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The third law of thermodynamics implies there can be no vacuum. But, wouldn't that imply that there always has to be at least one particle within an arbitrarily defined box? Can there be empty space?

I've been told the temperature "of a" vacuum is 0K, but doesn't temperature require the presence of energy? I.e. the internal energy of something. Isn't temperature always the temperature of (a collection of) particles and not of space itself? If there are no particles, how can there be temperature?

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