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Consider the Otto cycle for an ideal gas engine consisting of two quasistatic adiabatic and two quasistatic isochoric processes. The correct temperature-entropy (T-S) phase diagram for the cycle is

  • A.Temperature-entropy diagram: cycle 1 to 2 up, 2 to 3 along a rising curve, 3 to 4 down, 4 to 1 along a curve back
  • B.Temperature-entropy diagram: cycle traversed 1, 2, 3, 4 with 4 above 1 on the left, curved top and bottom edges, run in the opposite sense
  • C.Temperature-entropy diagram: rectangular cycle 1, 2, 3, 4 with straight horizontal and vertical sides, run clockwise
  • D.Temperature-entropy diagram: rectangular cycle with 4 above 1 on the left and 3 above 2 on the right, run anticlockwise

Answer: A

The correct T-S diagram is (A).

The Otto cycle is 1 to 2 adiabatic compression, 2 to 3 heat addition at constant volume, 3 to 4 adiabatic expansion and 4 to 1 heat rejection at constant volume. A quasistatic adiabatic process is isentropic, so 1 to 2 and 3 to 4 are vertical lines. At constant volume \(dS = C_{V}\,dT/T\), so T rises exponentially with S along 2 to 3 and falls along 4 to 1: curved lines rising to the right. An engine runs clockwise on a T-S diagram, taking in heat at the higher temperature.

Diagram (A) has exactly this shape and direction. (B) has the same shape traversed anticlockwise, which is a refrigerator, not an engine. (C) and (D) are rectangles, with horizontal isothermal sides, which describe the Carnot cycle, not the Otto cycle.

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