(b) Second law of thermodynamics

(c) Entropy

5. (20%) There are two chambers filled with air and butane in an insulated cylinder as shown in the figure below. The plate that separates the two chambers is rigid which means it does not move. Both chambers remain at the same temperature due to the heat transfer through the rigid plate. A piston is used to compress the air from the left side. Initially air is at 500 KPa, 200℃ with volume of 0.01 m3 and butane is at 1500 kPa, 200℃ with the volume of 0.01 m3. Air is compressed reversibly such that the temperature of the gases is raised to 225*C. Determine: (1) final pressure of both gases； (2) heat transferred between the chambers； (3) the work performed on air. All heat losses to the surroundings can be neglected. Assume air and butane to be ideal gases with constant specific heats.

6. (18%) A cycle consists of two iso-pressure process connected by two reversible adiabatic processes as shown in the figure below. Show that the thermnal efficiency of a reversible heat engine operating on such an air-standard cycle with constant  

(a) If A and B are symmetric then AB is symmetric.

(b) If AB = Bthen A = I.

(c) All geometric progressions

(d) If subspace V is orthogonal to W, then

(e) If A is invertible and B is singular, then A + B is invertible.

(a) Find an orthonormal basis for the column space of A.

(b) Write A as QR, where Q has orthonormal columns and R is upper triangular.

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