For a quasi-static adiabatic process (no heat transfer, Q=0) in an ideal monatomic gas, the First Law of Thermodynamics combined with the internal energy relation U = (3/2)nRT and the ideal gas law PV = nRT can be integrated to yield the invariant relation (T_f/T_i)^(3/2) · (V_f/V_i) = 1 between any two states connected by an adiabatic path. Applying this relation to the two adiabatic legs of a Carnot cycle (which share the same pair of isotherms at temperatures T1 and T2) and eliminating the temperature-ratio term shows that the ratio of volumes across each adiabatic leg is equal, establishing the general volume-ratio symmetry property of the Carnot cycle. This belongs to classical thermodynamics, specifically the theory of adiabatic processes and cyclic heat-engine analysis within the study of ideal gases.
For a quasi-static adiabatic process (no heat transfer, Q=0) in an ideal monatomic gas, the First Law of Thermodynamics combined with the internal energy relation U = (3/2)nRT and the ideal gas law P…