An ideal monatonic gas is confined in a cylinder by a spring-loaded piston of cross section `8.0xx10^(-3) m^(2).`initially the gas is at 300K and occupies a volume of `2.4xx10^(-3)m^(-3)` and the sprinng is in its relaxed state. The gas is heated by a small heater until the piston moves out slowly by 0.1m. Calculate the final tempreture of the gas. The force constant if the spring is `8000Nm^(-1)`,and the atmospheric pressure is `1.0xx10^(5)Nm^(-2)`. the cylinder and the piston are thermally insulated. The piston and the spring are massless and there is no friction between the piston and the cylinder. Neglect any heat-loss through the lead wires of the heater. The heat capacity of the heater coil is negligible. assume the spring to be massless

Explanation

Initially the spring is in its relaxed state, So, the pressure of the gas equals the atmospheric pressure, Initial pressure`=p_(1)=1.0xx10^(5)Nm^(-2).`
Final pressure `=p_(2)=p_(1)+(kx)/(A)`
`=1.0xx10^(5)Nm^(-2)+((8000Nm^(-1))((0.1m)))/(8.0xx10^(-3)m^(2)`
`=2.0xx10^(5)Nm^(-2)`
`Final volume =V_(2)=V_(1)+Ax`
`=2.4xx10^(-3)m^(3)+8.0xx10^(-3)m^(2)xx0.1m=3.2xx10^(-3)m^(3).`
`Using (p_(1)V_(1))/(T_(1))=(p_(2)V_(2))/(T_(2)),`
`T_(2)=(p_(2)V_(2))/(p_(1)V_(1))T_(1)`
`=((2.0xx10^(5)Nm^(-2))(3.2xx10^(-3)m^(3)))/((1.0xx10^(5)Nm^(-2))(2.4xx10^(-3)m^(3))xx300K`
`=800K.`