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By using Direct Numerical Simulations (DNS) of bubbly flows with passive scalars, we show a transition in the scalar spectra from a $k^{-5/3}$ to a $k^{-3}$ scaling with the wavenumber $k$, in contrast with those of single-phase isotropic turbulence. For cases with a mean scalar gradient in the horizontal direction, the scalar spectrum decays faster than $k^{-3}$ at high wavenumbers. While the $k^{-3}$ scaling is well established in the bubbly flow velocity spectrum, the scalar spectrum behaviour is not fully understood. We find that the transition length scale of the scalar spectra is comparable to or below the bubble diameter and decreases with the molecular diffusivity of the scalar in the liquid. We use DNS to compute the scalar spectra budget and show that the scalar fluctuations are produced by the mean scalar gradient at length scales above the bubble diameter, contrary to the velocity fluctuations. At length scales below the bubble diameter, the net scalar transfer scales as $k^{-1}$ inducing the $k^{-3}$ scaling of the scalar spectra. This finding is consistent with the hypothesis proposed by Dung et al. (2023) about the physical mechanism behind the $k^{-3}$ scaling. We also show dependencies of the bubble suspension's convective scalar diffusivity on the gas volume fraction and molecular diffusivity that differs based on the direction of the mean scalar gradient. For a mean scalar gradient in the vertical direction, we find and qualitatively explain a significant effect of the molecular diffusivity in the gas on the convective scalar diffusivity.