Question

Identify the equation typically used to describe the efficiency of a cyclic process.

• A. \( \eta = 1 - \frac{Q_{out}}{Q_{in}} \)
• B. \( PV = nRT \)
• C. \( \Delta G = \Delta H - T\Delta S \)
• D. \( F = ma \)

Hint:

Thermal Efficiency. For a heat engine cycle: Efficiency \(\eta\) = (Net Work Out) / (Heat In) = \(W_{net/Q_{in\). Using the first law (\(W_{net = Q_{in - Q_{out\)), \(\eta = (Q_{in - Q_{out)/Q_{in = 1 - Q_{out/Q_{in\).

Answer 1

The Correct Option is A

The thermal efficiency (\(\eta\)) of a cyclic process, particularly a heat engine, is defined as the ratio of the net work output (\(W_{net}\)) to the heat input (\(Q_{in}\) or \(Q_H\)) from the high-temperature reservoir.
$$ \eta = \frac{W_{net}}{Q_{in}} $$ According to the first law of thermodynamics for a cycle, the net work done equals the net heat transferred: \(W_{net} = Q_{net} = Q_{in} - Q_{out}\), where \(Q_{out}\) (or \(Q_C\)) is the heat rejected to the low-temperature reservoir (magnitude).
Substituting this into the efficiency definition: $$ \eta = \frac{Q_{in} - Q_{out}}{Q_{in}} = 1 - \frac{Q_{out}}{Q_{in}} $$ This equation relates efficiency to the heat absorbed and rejected during the cycle.
Option (1) represents this fundamental definition of thermal efficiency for a cyclic process operating as a heat engine.
Option (2) is the ideal gas law.
Option (3) relates Gibbs free energy, enthalpy, and entropy.
Option (4) is Newton's second law.