We present the result of a design study for X-Spec, a multi-beam, R=400-700 survey spectrometer covering 190-520 GHz under development for CCAT. It is designed to measure the bright atomic fine-structure and molecular rotational transitions that cool galaxies' interstellar gas, in particular, the 158 µm rest-frame [CII] transition, in thousands to tens of thousands of galaxies ranging from z=9 to z=3.5. With the wide bandwidth and multi-object capability, X-Spec / CCAT will be more powerful than ALMA for redshift-blind galaxy surveys and tomographic intensity mapping. X-Spec uses SuperSpec filterbank spectrometer technology with TiN KIDs described by Hailey-Dunsheath et al. in this conference. Because the density of sources is small, galaxy follow-up will be most efficient with a front-end steering unit which we have prototyped, also described in a separate paper (Chapman et al. in this conference). Our baseline instrument concept has 84 steered beams arrayed over the 1 degree CCAT field, each beam couples to 4 chips (2 bands x 2 polarizations) each chip with approximately 500 detectors, making a total of ∼170,000 KIDs in the full instrument. A direct imaging spectrometer (integral-field spectrometer) with a comparably-sized backend is also considered.