 Solution

For $\nu =1/2$, the gas-phase metallicity relation reduces to $Z=2y(1-M/M_0)$. Elimination of $M$ from mass conservation givesWith a fixed initial mass function, let $\kappa$ be the number of surviving low-mass tracer stars per unit formed stellar mass. Successive stars inherit the monotonically increasing gas-phase metallicity, so the cumulative and differential counts are respectivelywhere $Z_f=2y(1-M_f/M_0)$ is the abundance at the epoch when star formation stops. Above $Z_f$ the cumulative count is constant, and the differential count vanishes. In the complete gas-exhaustion limit, $Z_f\to 2y$ and $N_{\mathrm{total}}=2\kappa M_0$, so the normalized metallicity distribution function is $p_Z=1/(2y)$ on $0<Z<2y$; at a finite stopping epoch it is $1/Z_f$. The distribution is flat in linear metallicity. A logarithmic metallicity distribution instead satisfies $dN/d{\log }_{10}Z=(\ln 10)ZdN/dZ$ and is not flat.