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But you've already given yourself away: You say you can *derive* this
relation between energy and temperature, starting from the ideal
gas law. I absolutely agree. Which proves that the relation is not
a definition. You can't derive a definition. So you need some *other*
definition of temperature to determine what the T in the ideal gas
law stands for. Even in an upper-division course, I think this other
definition should be an operational one (how to make a thermometer).
Then, when students ask what temperature really *is*, even in an
introductory course, you need to tell the truth: it's a measure of
the willingness of an object to spontaneously give up energy. In many,
but not all cases, this "willingness" happens to be proportional to
how much energy the object already has.