PHY.3:5.1 - Bound a proposed engine without designing its mechanism

Question. A proposed cyclic machine takes 100 J from an ideal reservoir at 600 K and delivers 60 J as work. Its only other interaction is heat rejection to an ideal reservoir at 300 K. The machine returns to its initial state. Can a different mechanism make the proposal possible under these same conditions?

The physical preparation includes both reservoirs, the work store and restoration of the machine. It supplies no fuel, depleted auxiliary store or initially available nonthermal resource. The two temperatures are absolute thermodynamic temperatures. Energy balance gives 40 J rejected to the cold reservoir; that balance alone leaves the proposed output admissible.

Use the classical second-law account for these conditions. Its Clausius restriction excludes a composite process whose only net effect is transfer of heat from the colder reservoir to the hotter one. An admitted reversible reference between these reservoirs has eta_ref = 1 - 300/600 = 1/2 and can run backward. This is a theoretical reference; no claim of a finite-power reversible laboratory device is needed.

Choose cancellation of the work exchange. A backward reference consuming the proposed 60 J extracts 60 J from the cold reservoir and delivers 120 J to the hot reservoir. The compatible exchanges combine as follows; positive entries increase the named store.

The net process transfers 20 J from cold to hot with no other change. It violates the selected restriction. The argument uses only the proposed exchanges and cyclic condition, so changing the candidate’s internal mechanism cannot repair it within this class.

For a general positive requested work W, the backward reference requires Q_hot_ref=W/eta_ref. If W > eta_ref*Q_hot, then Q_hot_ref > Q_hot; canceling work again produces the prohibited cold-to-hot transfer. Therefore:

W <= eta_ref*Q_hot = (1 - T_cold/T_hot)*Q_hot.
For the stated inputs: W <= 50 J.

The practical result redirects the design question to work below this bound or to a change in the stated resources. It does not supply the design that attains a chosen value, its power or its operating cost.

Change the allowed physical operation. Suppose the proposed device may also receive 20 J of external work. A forward reversible reference producing 40 J from 80 J of hot-reservoir heat, together with routing those 20 J through the work store, can supply 60 J gross output. The net work produced is 40 J. The denominator and net exchange must now reflect the actual question; this construction supplies no engine producing 60 J net work from the original 100 J alone. If exactly 100 J must still be taken from the hot reservoir, an admitted additional transfer of 20 J from hot to cold accounts for the remainder. The total cold-reservoir gain is 60 J, and both laws permit the resulting exchanges. The changed resource condition opens a construction rather than altering the earlier bound.

For a microscopic proposal with initial correlations or a nonthermal auxiliary, use the physical account appropriate to those resources. A reservoir-only calculation leaves their contribution out. The source comparison in :11 identifies a current treatment; it does not prescribe the same generalized formula for every macroscopic device.