Impossibility of an engine working only by the First Law.

The idea

The first law is an energy audit: it says energy is conserved, not which way a process may run. A device could, in principle, take heat Q from a hot reservoir and convert it entirely into work W with ΔU = 0, satisfying ΔU = Q − W. The second law forbids this in a cyclic engine: no engine operating between two reservoirs can convert all absorbed heat into work, so efficiency is always below 1. The Kelvin–Planck statement captures this. A real engine must reject some heat Q_c to a cold sink, so W = Q_h − Q_c and efficiency = W/Q_h. This is why a single reservoir cannot drive a cyclic engine, and why 100% efficiency is unattainable.

What earns the marks

• State that the first law expresses conservation of energy and does not restrict the direction of energy transfer.
• Explain that a cyclic engine with one reservoir would need Q_c = 0, giving efficiency 1, which the second law forbids.
• Quote or apply the Kelvin–Planck statement: no cyclic device converts all absorbed heat into work.
• Use W = Q_h − Q_c and efficiency = W/Q_h to show that efficiency is always less than 1 when Q_c is positive.
• Distinguish a cyclic engine from a single one-off expansion, where all the heat could in principle become work.
• Recognise that the second law is an independent principle, not derivable from the first law alone.

Common mistakes and how to put them right

• Claiming the first law alone forbids a 100% efficient engine.

  Put it right: The first law permits it and the second law forbids it.

• Confusing efficiency with the fraction of heat rejected.

  Put it right: Efficiency is useful work divided by heat absorbed.

• Applying the Kelvin–Planck restriction to a non-cyclic process.

  Put it right: It applies to devices operating in a cycle.

• Omitting the cold reservoir when describing a heat engine.

  Put it right: A hot source, a cold sink and work output are all required.

Exam tips

• Name the law you are invoking explicitly, since questions often test whether you can separate the roles of the first and second laws.
• Write the energy balance W = Q_h − Q_c before substituting numbers, then form efficiency as W/Q_h.
• When asked why an engine cannot be 100% efficient, refer to the need to reject heat to a cold sink, not just to friction.
• Use clear reservoir labels (hot and cold) in diagrams so the direction of each heat transfer is unambiguous.