  Supernovae (SN) generate hot gas in the interstellar medium (ISM), help
setting the ISM structure and support the driving of outflows. It is important
to resolve the hot gas generation for galaxy formation simulations at solar
mass and sub-parsec resolution which realise individual supernova (SN)
explosions with ambient densities varying by several orders of magnitude in a
realistic multi-phase ISM. We test resolution requirements by simulating SN
blast waves at three metallicities ($Z = 0.01, 0.1$ and $1 Z_{\odot}$), six
densities and their respective equilibrium chemical compositions ($n=0.001$
cm$^{-3}$ - $100$ cm$^{-3}$), and four mass resolutions ($0.1$ - $100$
M$_{\odot}$), in three dimensions. We include non-equilibrium cooling and
chemistry, a homogenous interstellar radiation field, and shielding with a
modern pressure-energy smoothed particle hydrodynamics (SPH) method including
isotropic thermal conduction and a meshless-finite-mass (MFM) solver. We find
stronger resolution requirements for chemistry and hot phase generation than
for momentum generation. While at $10$ M$_{\odot}$ the radial momenta at the
end of the Sedov phase start converging, the hot phase generation and chemistry
require higher resolutions to represent the neutral to ionised hydrogen
fraction at the end of the Sedov phase correctly. Thermal conduction typically
reduces the hot phase by $0.2$ dex and has little impact on the chemical
composition. In general, our $1$, and $0.1$ M$_{\odot}$ results agree well with
previous numerical and analytic estimates. We conclude that for the thermal
energy injection SN model presented here resolutions higher than $10$
M$_{\odot}$ are required to model the chemistry, momentum and hot phase
generation in a multi-phase ISM.
