Question

Determine the work A done by an ideal gas during a closed cycle 1 → 4 → 3 → 2 → 1 shown in Fig. 59 if p₁ = 10⁵ Pa, p₀ =3 x 10⁵ Pa, p₂ =4 x10⁶ Pa, V₂ - V₁ = 10 l, and segments 4-3 and 2-1 of the cycle are parallel to the V-axis.

Answer 1

The cycle 1→ 4 → 3 → 2 → 1 is in fact equivalent to two simple cycles 1 → 0 → 2 → 1 and 0 → 4 → 3 → 0 (see Fig. 59). The work done by the gas is determined by the area of the corresponding cycle on the p-V diagram. In the first cycle the work is positive, while in the second cycle it is negative (the work is done on the gas). The work done in the first cycle can easily be calculated:

As regards the cycle 0 → 4 → 3 → 0, the triangle on the p-V diagram corresponding to it is similar to the triangle corresponding to the first cycle. Therefore, the work A₂ done in the second cycle will be

(The areas of similar triangles are to each other as the squares of the lengths of the corresponding elements, in our case, altitudes.) The total work A done during the cycle 1 → 4 → 3 → 2 → 1 will therefore be