The initial temperature and volume of a gas are 273 K and 6.00 x10 -4 m 3 . The

Question

The initial temperature and volume of a gas are 273 K and 6.00 x10-4m3. The initial and final amounts bywhich the spring is stretched are, respectively, 0.0800 and 0.1000m. The cross sectional area of the piston is 2.50 x10-3m2. What is the final temperature of thegas? NOTE: The ideal gas is confined to a cylinder by amassless piston that is attached to an ideal spring. Outside thecylinder is a vacuum. The cross-sectional area of the piston is A.The inital pressure, volume, and temperature of the gas are,respectively, P0 , V0 , andT0 , and the spring is initially stretched byan amount x0 with respect to its unstrained length. Thegas is heated, so that its final pressure, volume, and temperatureare Pf . Vf , and Tf and thespring is stretched by an amount xf with respect to itsunstrained length. Assume that x0 and xf arepositive variables.

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    the final temperature can be obtained fromthe relation    Tf = (Pf Vf/ Po Vo) To    the final temperature can be obtained fromthe relation    Tf = (Pf Vf/ Po Vo) To

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