Tetragonal crystals were formed at addition of yttrium oxide on concentration range from 2 to 4 mol. %. It was estimated the thermodynamic functions of these systems, in particular on values of free surface energy (Gs) and elastic energy (Ge) of crystals and their ability to respond on change of external conditions – temperature. It was shown for range of small sizes (size decreases at temperature growth) the concentration of dopants practically doesn’t influence on dynamic of change of Gs then as for range of particles size of 15-35 nm the concentration of dopant more significantly influence on the Gs. Analysis of dynamic of change of Gs and Ge values shows that for systems which have been formed at heat treatment before 700°C the change of these thermodynamics functions is bigger. In this time the dynamic of change of total Gs + Ge value for systems which was synthesized at different temperatures shows the influence on yttria concentration.