Question

The drawing shows an ideal gas confined to a cylinder by a massless piston that is attached to an ideal spring. Outside the cylinder is a vacuum. The cross-sectional area of the piston is A = 2.50 × 10^-3 m². The initial pressure, volume, and temperature of the gas are, respectively, P₀, V₀ = 6.00 × 10^-4 m³ and T₀ = 273 K, and the spring is initially stretched by an amount x₀ = 0.090 m with respect to its unstrained length. The gas is heated, so that its final pressure, volume, and temperature are P_f, V_f and T_f and the spring is stretched by an amount x_f = 0.12 m with respect to its unstrained length. What is the final temperature of the gas?

Answer 1

Initial pressure Po can be obtained from the stretiching force of spring.

Let k be the spring constant, then initial stretching force = k 0.09 N

Hence initial pressure Po = k 0.09 A   N/m2 ..............(1)

where A is area of piston

Similarly final pressure Pf = k 0.12 A   N/m2 ............ (2)

from eqn.(1) and eqn.(2), we get Pf = (4/3) Po

when the spring is stretched from 0.09 m to 0.12 m , change in volume is given by

= 0.03 A = 0.03 2.5 10-3 m3 = 0.075 10-3 m3

Final volume Vf = Vo + = 6.0 10-4 + 0.75 10-4 = 6.75 10-4 m3

For ideal gas , ( Po Vo / To ) = ( Pf Vf / Tf )

Hence , we get , Tf = To [ ( Pf Vf ) / (PoVo) ] = 273 (4/3) (6.75 / 6 ) = 409.5 K