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Reversible Adiabatic Paths do not Intersect

Let us assume any two adiabatic paths CA & CB which intersect at C. Let us also draw one isothermal P path which intersects CA and CB at A and B respectively. CA, AB and BC now constitute a cycle which is reversible. The work done on the system is the area included between them. During paths CA and CB, there is no heat transfer, the heat transfer takes place only the reversible isothermal path at a single temperature reservoir. So this cycle violates the 2nd law.

Thus this reversible cycle is impossible. So the starting assumption that two reversible adiabatic paths intersect is false. So two reversible adiabatic paths cannot intersect each other.