First-principle calculations have been carried out on the structural, electronic, elastic, and phonon properties of the full-Heusler alloys X2YAl (X = Co, Fe and Y = Cr, Sc). The calculations predict that the Fe2CrAl and Co2CrAl are half-metallic ferromagnets at the equilibrium lattice constant with a minority-spin energy gap of 0.2912 and 0.668 eV, respectively. Fe2ScAl exhibit a gap in the majority density of states, with a few states at the Fermi level and about 0.217 states eV(-1), unlike the other Heusler compounds; due to this, it is considered a false half metal, and Co2ScAl is considered a non-magnetic compound. The elastic constants were derived from the slopes of the acoustic branches in the phonon-dispersion curve. The calculated lattice constants, bulk modulus, and first-order pressure derivative of the bulk modulus are reported for the L2(1) structure and compared with previous values. Phonon-dispersion curves were obtained using the first-principle linear-response approach of the density-functional perturbation theory. The specific heat capacity at a constant volume C (V) of X2YAl (X = Co, Fe and Y = Cr, Sc) alloys is calculated and discussed.