= Finite battery charge reservoir
{title2=$Q_{\rm delivered}=\int I\,dt$}
A <battery>'s charge reservoir means a finite amount of <electric charge> can pass through the external <electrical circuit> before its usable chemical state is depleted. It does not mean a large net <electric charge> is sitting inside the casing. For one net cell <chemical reaction> transferring $n$ <Electrons> per reaction event, an ideal reaction inventory obeys
$$
Q_{\rm delivered}=\int_0^t I(t')\,dt'=nF\xi,\qquad
W_L=\int_0^t V_L(t')I(t')\,dt',
$$
where $F$ is the <Faraday constant> and $\xi$ the <extent of reaction> in <moles>, assuming all the <electric current> serves that reaction. At fixed <temperature> and <pressure>, its reversible <electromotive force> obeys $\Delta_rG=-nF\mathcal E$: a decrease in <Gibbs free energy> provides the maximum reversible electrical <work> per <mole>. The <electromotive force> also depends on composition; the <Daniell cell> illustrates this with <thermodynamic activity>.<Chemistry 2e by Paul Flowers and others/17.4 Potential, Free Energy, and Equilibrium>[§17.4, E° and ΔG°, and Potentials at Nonstandard Conditions: free energy, cell potential and composition]{ref}
As an original finite-reservoir exercise, stipulate usable $Q_{\max}=360\,\mathrm C$ and hold the preceding $3\,\mathrm V$, $1\,\Omega$, $4\,\Omega$ parameters constant until an abrupt cutoff. Then $t=Q_{\max}/I=600\,\mathrm s$, available electrical <work> is $\mathcal E Q_{\max}=1080\,\mathrm J$, load <work> is $V_LQ_{\max}=864\,\mathrm J$, and internal dissipation is $216\,\mathrm J$. This explicitly finite idealization predicts ten minutes, not an experimentally measured runtime. Real accessible charge depends on discharge conditions and the chosen terminal-voltage cutoff. Charge alone is also not an <energy>: converting it to delivered <work> requires the terminal <electric potential difference> throughout discharge.
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