= Solution
The <Kennicutt–Schmidt law> is the empirical relation
$$
\Sigma_{\rm SFR}=A\Sigma_g^N,
\qquad N\simeq1.4
$$
for disk-averaged total gas, with a nearly linear molecular-gas relation in many resolved observations. Atomic gas is mapped through the H I 21-cm line, molecular gas mainly through carbon-monoxide line emission and a CO-to-$\mathrm H_2$ conversion factor, and star formation through combinations of ultraviolet continuum, H-alpha recombination emission, and infrared dust emission. Inclination, dust attenuation, the <initial mass function>, tracer lifetimes, and conversion factors must be treated consistently.
The gas-depletion time $M_g/\dot M_*$ is typically of order a gigayear, whereas a giant molecular cloud has a dynamical or free-fall time of order a few megayears. Star formation is therefore inefficient per collapse time, commonly at the percent level, rather than converting an entire cloud in one free fall.
For the first <closed-box model of galactic chemical evolution>, neglect returned mass or absorb it into the definitions. Then
$$
\frac{dM_g}{dt}=-\dot M_*=-\frac{M_g}{\tau_*},
$$
and hence
$$
\boxed{M_g=M_{g0}e^{-t/\tau_*},
\qquad \dot M_*=\frac{M_{g0}}{\tau_*}e^{-t/\tau_*},
\qquad Z=y_Z\log\frac{M_{g0}}{M_g}=\frac{y_Zt}{\tau_*}.}
$$
Thus a region reaching $Z_\odot=0.014$ after an enrichment time $t_\odot$ with $y_Z=0.006$ has
$$
\boxed{\tau_*=\frac{y_Z}{Z_\odot}t_\odot\simeq0.43t_\odot,}
$$
about $4.3\,\mathrm{Gyr}$ for a fiducial $t_\odot\simeq10\,\mathrm{Gyr}$ enrichment age of the Galactic disk.
The second prescription implies $\dot M_*=M_g^2/(M_{g0}\widetilde\tau_*)$. Solving the gas-consumption equation gives
$$
\boxed{M_g=\frac{M_{g0}}{1+t/\widetilde\tau_*},
\qquad
\dot M_*=\frac{M_{g0}}{\widetilde\tau_*}
\left(1+\frac{t}{\widetilde\tau_*}\right)^{-2},
\qquad
Z=y_Z\log\left(1+\frac{t}{\widetilde\tau_*}\right).}
$$
Therefore
$$
\boxed{\widetilde\tau_*
=\frac{t_\odot}{e^{Z_\odot/y_Z}-1}
\simeq0.107t_\odot\simeq1.1\,\mathrm{Gyr}.}
$$
For long-lived stars, $dN$ is proportional to $dM_*=-dM_g$. In the exponential model,
$$
\frac{dN}{dZ}\propto
\dot M_*\frac{dt}{dZ}
=\frac{M_{g0}}{y_Z}e^{-Z/y_Z}.
$$
In the second model, $1+t/\widetilde\tau_*=e^{Z/y_Z}$; multiplying its star-formation rate by $dt/dZ$ gives exactly the same result:
$$
\boxed{\frac{dN}{dZ}\propto e^{-Z/y_Z}.}
$$
The metallicity distribution is fixed by the closed-box relation $M_g/M_{g0}=e^{-Z/y_Z}$ and is independent of the star-formation history. Merely changing the time law therefore does not cure the <G-dwarf problem>; gas inflow, outflow, variable yields, or selection effects must alter the closed-box assumptions.
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