  A simple analytic description is provided of the rate of energy deposition by
$\beta$-decay electrons in the homologously expanding radioactive plasma
ejected in neutron star mergers, valid for a wide range of ejecta parameters --
initial entropy, electron fraction $\{s_0,Y_e\}$ and density $\rho t^3$. The
formulae are derived using detailed numerical calculations following the
time-dependent composition and $\beta$-decay emission spectra (including the
effect of delayed deposition). The deposition efficiency depends mainly on
$\rho t^3$ and only weakly on $\{s_0,Y_e\}$. The time $t_e$ at which the ratio
between the rates of electron energy deposition and energy production drops to
$1-e^{-1}$, is given by $t_e=t_{0e}\Big(\frac{\rho t^3}{0.5(\rho
t^3)_0}\Big)^a$, where $(\rho t^3)_0=\frac{0.05M_{\odot}}{4\pi(0.2c)^3}$,
$t_{0e}(s_0,Y_e)\approx17$ days and $0.4\le a(s_0,Y_e)\le0.5$. The fractional
uncertainty in $t_e$ due to nuclear physics uncertainties is $\approx10\%$. The
result $a\le0.5$ reflects the fact that the characteristic $\beta$-decay
electron energies do not decrease with time (largely due to "inverted decay
chains" in which a slowly-decaying isotope decays to a rapidly-decaying isotope
with higher end-point energy). We provide an analytic approximation for the
time-dependent electron energy deposition rate, reproducing the numerical
results to better than $50\%$ (typically $<30\%$, well within the energy
production rate uncertainty due to nuclear physics uncertainties) over a 3-4
orders-of-magnitude deposition rate decrease with time. Our results may be
easily incorporated in calculations of kilonovae light curves (with general
density and composition structures), eliminating the need to numerically follow
the time-dependent electron spectra. Identifying $t_e$, e.g. in the bolometric
light curve, will constrain the (properly averaged) ejecta $\rho t^3$.
