Arnold Neumaier's comment about statistical mechanics is correct, but here's how you can prove it using just thermodynamics. Let's imagine two bodies at different temperatures in contact with one another. Let's say that body 1 transfers a small amount of heat Q to body 2. Body 1's entropy changes by −Q/T1, and body 2's entropy changes by Q/T2, so the total entropy change is
Q(1T2−1T1).
This total entropy change must be positive (according to the second law), so if
1/T1>1/T2 then
Q has to be negative, meaning that body 2 can transfer heat to body 1 rather than the other way around. It's the sign of
1T2−1T1 that determines the direction that heat can flow.
Now let's say that T1<0 and T2>0. Now it's clear that 1T2−1T1>0 since both 1/T2 and −1/T1 are positive. This means that body 1 (with a negative temperature) can transfer heat to body 2 (with a positive temperature), but not the other way around. In this sense body 1 is "hotter" than body 2.
This post imported from StackExchange Physics at 2014-04-04 16:14 (UCT), posted by SE-user Nathaniel