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We use ALMA observations of four sub-millimetre galaxies (SMGs) at $z\sim2-3$to investigate the spatially resolved properties of the inter-stellar medium(ISM) at scales of 1--5 kpc (0.1--0.6$''$). The velocity fields of our sources,traced by the $^{12}$CO($J$=3-2) emission, are consistent with disk rotation tofirst order, implying average dynamical masses of$\sim$3$\times10^{11}$M$_{\odot}$ within two half-light radii. Through aBayesian approach we investigate the uncertainties inherent to dynamicallyconstraining total gas masses. We explore the covariance between the stellarmass-to-light ratio and CO-to-H$_{2}$ conversion factor, $\alpha_{\rm CO}$,finding values of $\alpha_{\rm CO}=1.1^{+0.8}_{-0.7}$ for dark matter fractionsof 15 \%. We show that the resolved spatial distribution of the gas and dustcontinuum can be uncorrelated to the stellar emission, challenging energybalance assumptions in global SED fitting. Through a stacking analysis of theresolved radial profiles of the CO(3-2), stellar and dust continuum emission inSMG samples, we find that the cool molecular gas emission in these sources(radii $\sim$5--14 kpc) is clearly more extended than the rest-frame $\sim$250$\mu$m dust continuum by a factor $>2$. We propose that assuming a constantdust-to-gas ratio, this apparent difference in sizes can be explained bytemperature and optical-depth gradients alone. Our results suggest that cautionmust be exercised when extrapolating morphological properties of dust continuumobservations to conclusions about the molecular gas phase of the ISM.

Resolving the ISM at the peak of cosmic star formation with ALMA - The distribution of CO and dust continuum in z~2.5 sub-millimetre galaxies