 Solution

The closed-box model of galactic chemical evolution assumes a fixed total baryonic mass, no gas inflow or outflow, instantaneous and homogeneous mixing, a constant stellar yield, a fixed initial mass function, and usually the instantaneous recycling approximation. It predicts $Z=y\log (1/\mu )$ for gas fraction $\mu$, but real galaxies violate these assumptions.

Evidence for galactic outflows includes blueshifted absorption, broad or split emission lines, extraplanar ionized and molecular gas, X-ray bubbles, and metal-enriched circumgalactic material. The low baryon fractions and low effective yields of dwarf galaxies, together with the galaxy mass--metallicity relation, provide indirect evidence for preferential gas and metal loss. Sustained star formation for longer than a closed reservoir's depletion time, low-metallicity high-velocity clouds, metallicity dilution during starbursts, the G-dwarf problem, and circumgalactic or intergalactic accretion signatures imply continuing galactic gas inflow.

Let $f(t)=A{e}^{-t/\tau }$ be the inflow rate, let the outflow rate be $\lambda \psi$, and let $R$ be the promptly returned fraction. The total baryonic, gas, and metal masses obeyThe first metal term is newly synthesized material; the second locks existing metals into long-lived stars and removes them in an ambient-composition wind. Expanding the final derivative and substituting ${\dot{M}}_g$ cancels those common terms, leaving

The observed Kennicutt–Schmidt law relates star-formation and gas surface densities approximately by ${\Sigma }_{\mathrm{SFR}}\propto {\Sigma }_g^{1.4}$, with a nearly linear relation to molecular gas over many resolved regimes. For the simple integrated model requested here, take a constant depletion coefficient $S$ so thatThen, with $\alpha =S(1-R+\lambda )$,The integrating factor $e^{\alpha t}$ gives, for $\alpha \ne 1/\tau$,At resonance, $\alpha =1/\tau$, the continuous limit isThe first term is depletion of the initial reservoir; the second is gas supplied by the exponentially declining inflow and subsequently consumed or expelled.