Active transport is a biological process in which substances are moved across cell membranes against their concentration gradient, meaning from an area of lower concentration to an area of higher concentration. This process requires energy, typically in the form of adenosine triphosphate (ATP), because it is opposing the natural flow of diffusion.
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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 Alberts et al.: carrier proteins and active membrane transport.
For the neutral 8:2 example, B-to-A transfer costs per mole. As an explicitly assumed bookkeeping example, suppose a tightly coupled cycle supplies a downhill free energy change of per mole of transferred solute. The total is then . 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.