Rational tangle replacements and knot Floer homology

From the link Floer complex of a link $K$, we extract a lower bound $t_q'(K)$ for the rational unknotting number of $K$ (i.e. the minimum number of rational replacements required to unknot $K$). Moreover, we show that the torsion obstruction $t_q(K)=\hat{t}(K)$ from an earlier paper of Alishahi and the author is a lower bound for the proper rational unknotting number. Moreover, $t_q(K\#K')=\max\{t_q(K),t_q(K')\}$ and $t'_q(K\#K')=\max\{t'_q(K),t'_q(K')\}$. For the torus knot $K=T_{p,pk+1}$ we compute $t'_q(K)=\lfloor p/2\rfloor$ and $t_q(K)=p-1$.