  We study the Clifford type inequality for a particular type of curves
$C_{2,2,5}$, which are contained in smooth quintic threefolds. This allows us
to prove some stronger Bogomolov-Gieseker type inequalities for Chern
characters of stable sheaves and tilt-stable objects on smooth quintic
threefolds. Employing the previous framework by Bayer, Bertram, Macr\`i,
Stellari and Toda, we construct an open subset of stability conditions on every
smooth quintic threefold in $\mathbf{P}^4_{\mathbb C}$.
