Classification of idealized thermodynamic gas power cycles:

• Dual cycle supplies heat partly at constant volume and partly at constant pressure, and rejects heat at constant volume.
• Otto cycle supplies heat at constant volume and rejects heat at constant volume.
• The Atkinson cycle (invented by James Atkinson in 1882) is an ideal thermodynamic cycle where heat addition occurs at constant volume, expansion continues down to atmospheric pressure (longer expansion stroke than compression stroke), and heat rejection takes place at constant pressure.
• Diesel cycle supplies heat at constant pressure and rejects heat at constant volume.

Thermodynamic efficiency of reversible Carnot heat engines:

• $\frac{T_1}{T_1 - T_2}$ represents the coefficient of performance (COP) of a Carnot heat pump, not engine efficiency.
• The thermal efficiency $\eta$ of a reversible Carnot heat engine operating between source temperature $T_1$ (higher temperature) and sink temperature $T_2$ (lower temperature) is given by: $\eta = \frac{W}{Q_1} = \frac{Q_1 - Q_2}{Q_1} = \frac{T_1 - T_2}{T_1} = 1 - \frac{T_2}{T_1}$.
• $\frac{T_2(T_1 - T_2)}{T_1(T_1 - T_2)}$ reduces algebraically to $T_2/T_1$, which is the fraction of heat rejected, not efficiency.
• $\frac{T_1 T_2}{T_1 - T_2}$ is dimensionally incorrect.

State functions and thermodynamic properties of ideal gases:

• Pressure alone does not dictate internal energy; at constant temperature, isothermal pressure changes cause zero change in ideal gas internal energy.
• Volume alone does not dictate internal energy; free expansion into vacuum produces no change in temperature or internal energy.
• According to Joule's law of ideal gases, the internal energy $U$ of an ideal gas depends solely and exclusively on its absolute temperature: $U = f(T)$, with $dU = m c_v dT$. Since intermolecular attractive forces are negligible in ideal gas models, changing volume or pressure without changing temperature does not alter molecular kinetic energy.
• For real gases, $U$ depends on $T$ and $v$, but for an ideal gas it is strictly a function of absolute temperature alone.

Core physical properties established by the four laws of thermodynamics:

• Zeroth law of thermodynamics defines the fundamental concept and empirical measurement of Temperature (thermal equilibrium, matching II).
• First law of thermodynamics introduces the conservation of energy and defines the state function Internal Energy ($dQ = dU + dW$, matching I).
• Second law of thermodynamics introduces the direction of natural spontaneous processes and defines the state property Entropy ($dS \ge dQ/T$, matching IV).
• Third law of thermodynamics (Nernst heat theorem) defines the absolute zero of entropy at Absolute zero temperature ($0\text{ K}$, matching III).