Here we learn that the compressible behavior of an ideal gas can be put to use in the design of a…

Here we learn that the compressible behavior of an ideal gas
can be put to use in the design of a spring. Consider a cylinder filled with an
ideal gas (cP, cv). In the center of this cylinder there
is a piston, which is pushed and displaced to the axial distance x by the
external force F = A(P1 _ P2), where A is the area
of the piston face, P1 is the pressure on the compressed side of the
piston, and P2 is the pressure on the expanded aside. Find the relationship
between F and x for two different designs:

(a) The process executed by the entire ideal gas system is
reversible and isothermal.
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Here we learn that the compressible behavior of an ideal gas
can be put to use in the design of a spring. Consider a cylinder filled with an
ideal gas (cP, cv). In the center of this cylinder there
is a piston, which is pushed and displaced to the axial distance x by the
external force F = A(P1 _ P2), where A is the area
of the piston face, P1 is the pressure on the compressed side of the
piston, and P2 is the pressure on the expanded aside. Find the relationship
between F and x for two different designs:

(a) The process executed by the entire ideal gas system is
reversible and isothermal. Show that F = kT x, where the kT
factor increases monotonically with x. Determine the form of the kT(x)
function in the limit of small displacements, x _
V0_A, where V0 is the initial size of the gas
volume on either side of the piston.

(b) The process executed by the entire ideal-gas system is
reversible and adiabatic. Invoke from the start the limit of small
displacements, show that F = kSx, and that the factor kS
increases as x increases. Compare the two designs, (a) and (b): which air
spring is stiffer?

(c) In both designs, the work transferred to the system is
Fx. Where does this amount of energy reside at the end of the process in design
(a) and in design (b)?

(d) Imagine that after the process of deforming the spring
the piston returns to its original position quasi-statically. What is the work
transfer from the ideal-gas system to the external entity that opposes by force
the return motion of the piston?

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