  A roughly constant temperature over a wide range of densities is maintained
in molecular clouds through radiative heating and cooling. An isothermal
equation of state is therefore frequently employed in molecular cloud
simulations. However, the dynamical processes in molecular clouds include shock
waves, expansion waves, cooling induced collapse and baroclinic vorticity, all
incompatible with the assumption of a purely isothermal flow. Here, we
incorporate an energy equation including all the important heating and cooling
rates and a simple chemical network into simulations of three-dimensional,
hydrodynamic, decaying turbulence. This allows us to test the accuracy of the
isothermal assumption by directly comparing a model run with the modified
energy equation to an isothermal model. We compute an extreme case in which the
initial turbulence is sufficiently strong to dissociate much of the gas and
alter the specific heat ratio. The molecules then reform as the turbulence
weakens. We track the true specific heat ratio as well as its effective value.
We analyse power spectra, vorticity and shock structures, and discuss scaling
laws for decaying turbulence. We derive some limitations to the isothermal
approximation for simulations of the interstellar medium using simple
projection techniques. Overall, even given the extreme conditions, we find that
an isothermal flow provides an adequate physical and observational description
of many properties. The main exceptions revealed here concern behaviour
directly related to the high temperature zones behind the shock waves.
