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Not all processes are amenable to decomposition into iso steps. Even if the process could be so treated, the heat and work effects would be dependent upon the path chosen. An example of a process restraint that must be treated in full is given below:

Example (a): A frictionless, massless piston is held in place inside an air-containing cylinder by a spring attached to a rigid wall. Heat is added until the volume is doubled. Find the changes in the other properties and the heat and work required to effect the change of state

The process is diagrammed below:

Heat is added to the air inside the cylinder until the volume is 0.2 m3 . The changes in all other properties and the heat and work are to be determined.

From the conditions given in the diagram, the number of moles of air in the cylinder is:

1 8.314-x 273K

The p-V path during the process is obtained from a force balance on the piston: P1A = k(xo - X1) initially and pA = k(xo - x) during expansion. Subtracting yields: p - p1 = (k/A)(x1 - x). Eliminating x using V - V1 = A(x1 - x) yields the p-V relation:

Substituting into Eq (3.1):

V2pdV = pj(V2 — Vi) + y2i-^- |(V2 — Vi)2 = 105 x0.1 + 0.5 x2x106(0.1)2 = 2x104 J

From the force balance on the piston: p2 = 0.1 + 2(0.2 - 0.1) = 0.3 MPa

From the ideal gas law

The increase in internal energy is:

AU = nCV(T2 - T1) = 4.41x ^ x8.314(1683 - 273) = 1.3x 105 J

From the 1s law:

Q = AU + W = 1.5x105 J AS = 4.41 x 8.314[(% )ln(1638/273) + ln(0.2/0.1)] = 190J/K

Example (b) Thermal equilibration of 2 solids - show that this process results in an entropy rise n =

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