A time-dependent solution for neutral-plasma isothermal expansions into a vacuum in a special-transformation coordinate system is obtained. In this new coordinate system, the special self-similar solutions of it were given by Huang and co-workers (Appl. Phys. Lett. 92, 031501). Combining the time-dependent solution and the quasi-linear increase of the electron density due to the hot-electron recirculation, an analytic model is proposed to reveal the influence of the hot-electron recirculation on the increase of electric field and on the acceleration of ions of different masses and charges.