An ideal gas is confined to a cylinder by a massless piston that is attached to

Question

An ideal gas is confined to a cylinder by a massless piston that is attached to an ideal spring. Outside the cylinder is vacuum. The cross-sectional area of the piston is A. the intail pressure, volume, and temperature of the gas are , respectively, P0, V0, and T0, and the spring is intially strectched by the amount x0 with respect to its unrestrained length. The gas is heated, so that its final pressure, volume, and temperature are Pf,Vf, and Tf and the spring is strectched by an amount xf with respect to unstrained length. Assume the x0 and xf are positive variables. (a) What is the realation between the magnitude of the force required to stretch an ideal spring and the amount of the stretch with respect to the unrestrained length of the spring? (b) What are the magnitudes, Fo and Ff, of teh forces thatthe intial and final pressures apply to the piston and, hence, to the spring? Express your answers in terms of the pressures and the cross-sectional area of teh piston. (c) According to the ideal gas law, how are the intial pressure, volume, and temperature related to the final pressure, volume, and temperature? (d) How is the final volume related to teh intial volume, the cross-sectional area of the piston, and the intial and final amounts by which the spring is stretched? Account for your answer

Explanation / Answer

(a)    the force Fapplied that mustbe applied to stretch an ideal spring by an amount x with respectto its unstrained length is    given by      Fapplied = k x    where k is the spring constant (b)    pressure is the magnitude of the forceapplied perpendicularly to a surface divided by the area of thesurface thus, the    magnitudes of the forces that the initialand final pressures apply to the piston (and, therefore, to thespring) are given by    Fo = Po A    Ff = PfA     (c)    the ideal gas law is    PV = nRT    since the number of moles is constant, thisequation can be written as    (P V / T) = n R    thus, the value of (P V / T) isthe same initially and finally, and we can write    Po Vo / To= Pf Vf / Tf (d)    the final volume is the initial volume plusthe amount by which the volume increases as the springstretches    the increased volume due to the additionalstretching is    A (xf - xo)    so we get that    Vf = Vo + A(xf - xo)    the final temperature canb be obtainedfrom    Tf = (Pf Vf/ Po Vo) To (d)    the final volume is the initial volume plusthe amount by which the volume increases as the springstretches    the increased volume due to the additionalstretching is    A (xf - xo)    so we get that    Vf = Vo + A(xf - xo)    the final temperature canb be obtainedfrom    Tf = (Pf Vf/ Po Vo) To

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