The sketch below shows a curve in p-V coordinates defined by the polytropic equa

Question

The sketch below shows a curve in p-V coordinates defined by the polytropic equation pVn = c, where n and c are positive constants. Verify the following related results: Note: the pdV integral, where p is the pressure and V is the volume, often occurs in the analysis of a thermodynamic system, and the polytropic equation, pVn = c, often describes the relationship between p and V when the analysis involves a gas. Thus, the importance of understanding the derivation of the previous equations cannot be overstated. Figure 6-1: Polytropic curve connecting thermodynamic states 1 and 2

Explanation / Answer

pVn = Cons.

=> p1V1n = p2V2n

=> lnp1 + nlnV1 = lnp2 + nlnV2

=>n[lnV1 - lnV2] = lnp2 - lnp1

=> n = ln(p2/p1)/ln(V1/V2) or n = ln(p1/p2)/ln(V2/V1)

pVn = COnst.

=> npVn-1dV + Vndp = 0

=> dV = -Vndp/npVn-1

Substituting, value of dV, we get the other results.

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