Question

An ideal monoatomic gas is confined in a cylinder by a
spring-loaded piston of cross-section 8.0 x 10$^{-3} m^2$. Initially
the gas is at 300 K and occupies a volume of 2.4 x 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 8000 N/m, and the atmospheric pressure
$2.0 \times 1^5 NM^{-2}$. The cylinder and the piston are thermally
insulated. The piston is massless 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.

Answer 1

Final pressure = $p_0 +\frac{kx}{A}$
$ \, \, \, \, \, \, \, =1.0 \times 10^5 + \frac{(8000)(0.1)}{8 \times 10^{-3}} = 2 \times 10^5 N/m^2$
Final volume = $2.4 \times 10^{-3} + (0.1) (8 \times 10^{-13})$
$\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, =3.2 \times 10^{-3} m^3 $
Applying, $ \, \, \, \, \, \frac{p_iV_i}{T_i} = \frac{p_f V_f}{T_f}$
we have, $T_f =\bigg(\frac{p_f V_f}{p_iV_i}\bigg)T_i$
$ \, \, \, \, \, \, \, \, \, =\frac{(2 \times 10^5)(3.2 \times 10^{-3})}{(1 \times 1^5)(2.4 \times 10^{-3})} \times 300 =800 K$
Heat supplied Q =$W_{gas} + \delta U$
$=P_0 (\Delta V)+\frac{1}{2} kx^2 +nC_v \Delta T \, \, \, \, \, \, \bigg(as \, n=\frac{p_iV_i}{RT+i}\bigg) $
=$(10)^5 (3.2-2.4) \times 10^{-3}+\frac{1}{2} \times 8000 \times (0.1)^2$
$+\frac{10^5 \times 2.4 \times 10^{-3}}{8.31 \times 300} \times \frac{3}{2} \times 8.31 \times (800-300)$
=80+40+600=720 J