  The controversial existence of negative temperatures has stirred interesting
debates that have reached the foundations of thermodynamics, including
questions on the second law, the Carnot efficiency and the statistical
definition of entropy. Here we show that for systems interacting with an
external field, negative temperatures arise from an energy mis-attribution in
which the interaction energy with the field is treated as a form of internal
energy. We discuss how negative temperatures are avoided when using a proper
thermodynamic formalism, which accounts for the intensive and extensive
variables associated to the external field. We use the paramagnetic system and
a perfect gas in a gravitational field to illustrate these ideas.
Considerations about the isothermal and adiabatic work done by the field or the
system also shed light on the inconsistency of super-Carnot efficiencies.
