The atomic dipole polarizability, $alpha$, and the van der Waals (vdW) radius, $R_{rm vdW}$, are two key quantities to describe vdW interactions between atoms in molecules and materials. Until now, they have been determined independently and separately from each other. Here, we derive the quantum-mechanical relation $R_{rm vdW} = const. timesalpha^{1/7}$ which is markedly different from the common assumption $R_{rm vdW} propto alpha^{1/3}$ based on a classical picture of hard-sphere atoms. As shown for 72 chemical elements between hydrogen and uranium, the obtained formula can be used as a unified definition of the vdW radius solely in terms of the atomic polarizability. For vdW-bonded heteronuclear dimers consisting of atoms $A$ and $B$, the combination rule $alpha = (alpha_A + alpha_B)/2$ provides a remarkably accurate way to calculate their equilibrium interatomic distance. The revealed scaling law allows to reduce the empiricism and improve the accuracy of interatomic vdW potentials, at the same time suggesting the existence of a non-trivial relation between length and volume in quantum systems.
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