An ideal gas is contained in a cylinder equipped with a frictionless, nonleaking

Question

An ideal gas is contained in a cylinder equipped with a frictionless, nonleaking piston of area A. When the pressure is atmospheric P0, the piston face is at a distance l from from the closed end. The gas is compressed by moving the piston a distance x. Calculate the spring constant, or force constant, F/x, under (a) isothermal conditions and (b) under adiabatic conditions. (c) In what respect is a gas cushion superior to a steel spring? An ideal gas is contained in a cylinder equipped with a frictionless, nonleaking piston of area A. When the pressure is atmospheric P0, the piston face is at a distance l from from the closed end. The gas is compressed by moving the piston a distance x. Calculate the spring constant, or force constant, F/x, under (a) isothermal conditions and (b) under adiabatic conditions. (c) In what respect is a gas cushion superior to a steel spring? An ideal gas is contained in a cylinder equipped with a frictionless, nonleaking piston of area A. When the pressure is atmospheric P0, the piston face is at a distance l from from the closed end. The gas is compressed by moving the piston a distance x. Calculate the spring constant, or force constant, F/x, under (a) isothermal conditions and (b) under adiabatic conditions. (c) In what respect is a gas cushion superior to a steel spring?

Explanation / Answer

(a) Under isothermal condition we know that
PV = Constant
Initial pressure is P0 and Volume = Al
Final volume = A(l-x)
Therfore final pressure
Pfinal = PoAl /(A(l-x)) = Pol / (l-x)
Now for spring constant = Force / deflection =( Pfinal A - Po A )/x
(b) In this case we can calcualte final pressure
PVn = C where n is adiabatic constant
therfore
Pinitial / Pfinal   =( Vfinal /Vinitial )n
Therefore we can calcualte final pressure as we know both volume
Pfinal = Po(Al/A(l-x))n  = Po(l/(l-x))n
hence now we put in the equation of spring constant then calcualte the value for spring constant.
Now for spring constant = Force / deflection =( Pfinal A - Po A )/x
(c) In gas cushion the damping properties are better than steel spring which lead to better service and longer life of
product.

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