Grain surface chemistry is key to the composition of protoplanetary disks around young stars. The temperature of grains depends on their size. We evaluate the impact of this temperature dependence on the disk chemistry. We model a moderately massive disk with 16 different grain sizes. We use POLARIS to calculate the dust grain temperatures and the local UV flux. We model the chemistry using the 3-phase astrochemical code NAUTILUS. Photoprocesses are handled using frequency-dependent cross-sections, and a new method to account for self and mutual shielding. The multi-grain model outputs are compared to those of single-grain size models (0.1 $mu$m), with two different assumptions for their equivalent temperature. We find that the Langmuir-Hinshelwood (LH) mechanism at equilibrium temperature is not efficient to form H$_2$ at 3-4 scale heights ($H$), and adopt a parametric fit to a stochastic method to model H$_2$ formation instead. We find the molecular layer composition (1-3 $H$) to depend on the amount of remaining H atoms. Differences in molecular surface densities between single and multi-grain models are mostly due to what occurs above 1.5 $H$. At 100 au, models with colder grains produce H$_2$O and CH$_4$ ices in the midplane, and warmer ones produce more CO$_2$ ices, both allowing efficient depletion of C and O as soon as CO sticks on grain surfaces. Complex organic molecules (COMs) production is enhanced by the presence of warmer grains in the multi-grain models. Using a single grain model mimicking grain growth and dust settling fails to reproduce the complexity of gas-grain chemistry. Chemical models with a single grain size are sensitive to the adopted grain temperature, and cannot account for all expected effects. A spatial spread of the snowlines is expected to result from the ranges in grain temperature. The amplitude of the effects will depend on the dust disk mass.
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