We show that the interaction of polar alkali-metal dimers in the quintet spin state leads to the formation of a deeply bound reaction complex. The reaction complex can decompose adiabatically into homonuclear alkali-metal dimers (for all molecules except KRb) and into alkali-metal trimers (for all molecules). We show that there are no barriers for these chemical reactions. This means that all alkali-metal dimers in the ${a}^{3}{\ensuremath{\Sigma}}^{+}$ state are chemically unstable at ultracold temperature, and the use of an optical lattice to segregate the molecules and suppress losses may be necessary. In addition, we calculate the minimum-energy path for the chemical reactions of alkali-metal hydrides. We find that the reaction of two molecules is accelerated by a strong attraction between the alkali-metal atoms, leading to a barrierless process that produces hydrogen atoms with large kinetic energy. We discuss the unique features of the chemical reactions of ultracold alkali-metal dimers in the ${a}^{3}{\ensuremath{\Sigma}}^{+}$ electronic state.
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