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 tenoerature of the gas. The force constant if the spring is `8000Nm^(-1)`,and the atmospheric pressure is `1.0xx10^(5)Nm^(-2).Hte cylinder and the piston are thermally insylated. 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. `
`

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.`