= Active transport
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<Active transport> moves a substance uphill in its <electrochemical gradient> by coupling the transfer to a downhill process. Some <carrier proteins> use <ATP> <hydrolysis>; others couple uphill transfer of one <solute> to downhill transfer of another <solute>, often an <ion>. The latter mechanism uses a stored <electrochemical gradient> and need not consume <ATP> directly at that <carrier protein>. Maintaining the driving <electrochemical gradient> still requires a source of <energy>. See https://www.ncbi.nlm.nih.gov/books/NBK26896/[Alberts et al.: carrier proteins and active membrane transport].
For the neutral 8:2 example, B-to-A transfer costs $RT\ln4\simeq1.38629RT$ per mole. As an explicitly assumed bookkeeping example, suppose a tightly coupled cycle supplies a downhill <free energy> change of $-2RT$ per mole of transferred <solute>. The total is then $\Delta G_{\mathrm{cycle}}=RT\ln4-2RT\simeq-0.61371RT$. This makes forward cycling energetically possible under those assumptions; it specifies neither a molecular mechanism nor a rate. Merely adding a <carrier protein> cannot supply the missing <free energy>. Conversely, as the transported <concentration> ratio rises, the required uphill cost rises, so a fixed driving process has a finite thermodynamic limit.
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