A semi$-$batch reactor initially contains a volume $V_0(m^3)$ of a solvent containing $N_{A0}$ moles of $A($or at the initial concentration of $C_{A0}).$ Reactant A decomposes by a second order reaction. The reaction mixture flows out of the reactor at a flow rate of $v\frac{m^3}{s}.$ Derive $($but do not solve$)$ the appropriate expression for $N_A$ as a function of time.