Purely kinetic κ-essence models in which the Lagrangian contains only a kinetic factor and does not depend explicitly on the field itself are considered.and a theoretical constraint is obtained:Fx=F0α-3.Under this theoretical constraint,we discuss a kind of purely k-essence with form F(X)=-(1+2Xn)1/2n,which can be considered as the generalized tachyon field,and find that this kind of κ-essence is not likely a candidate of dark energy to describe the present accelerated expansion of the Universe.This is contrary to a previous suggestion that κ-essence with such a form may be used to describe phantom cosmologies.