An ideal monoatomic gas is confined in a cylinder by a spring-located position of cross-section `8.0 xx 10^(-3)m^(2)`. Initially has the gas is at `300K` and occupies a volume of `2.4xx 10^(-3)m^(3)` and the spring is in its relaxed (unstretched, uncompressed)state. The gas is heated by a small electric heater until the piston moves out slowly by `0.1`m. Calculate the final temperature of the gas and the heat supplied (in joules) by the heater. The force constant of the spring is `8000N//m`, and the pressure `1.0 xx 10^(5)Nm^(-2)`. The cylinder and the piston are thermally insulated. The piston is massles and there is no friction between the piston and the cylinder . Neglect heat loss through the lead wires of the heater.The heat capacity of the heater coil is negligible . Assume the spring to the 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.`