  The collapse of the primordial gas in the density regime $\sim
10^{8}\hbox{--}10^{10}$ cm$^{-3}$ is controlled by the three-body $\rm H_2$
formation process, in which the gas can cool faster than free-fall time
$\hbox{--}$ a condition proposed as the chemothermal instability. We
investigate how the heating and cooling rates are affected during the rapid
transformation of atomic to molecular hydrogen. With a detailed study of the
heating and cooling balance in a 3D simulation of Pop~III collapse, we follow
the chemical and thermal evolution of the primordial gas in two dark matter
minihaloes. The inclusion of sink particles in modified Gadget-2 smoothed
particle hydrodynamics code allows us to investigate the long term evolution of
the disk that fragments into several clumps. We find that the sum of all the
cooling rates is less than the total heating rate after including the
contribution from the compressional heating ($pdV$). The increasing cooling
rate during the rapid increase of the molecular fraction is offset by the
unavoidable heating due to gas contraction. We conclude that fragmentation
occurs because $\rm H_2$ cooling, the heating due to $\rm H_2$ formation and
compressional heating together set a density and temperature structure in the
disk that favors fragmentation, not the chemothermal instability.
