Question

An ideal monoatomic gas is confined in a cylinder by a spring-loaded piston of area of cross-section $8\times {10}^{-3}{m}^2$. Initially, the gas is at 300 K and occupies a volume of $2.4\times {10}^{-3}{m}^3$ and the spring is in its relaxed state as shown in figure. The gas is heated by a small electric heater until the piston moves out slowly by 0.1 m. The force constant of the spring is 8000 N $m^{-1}$, atmospheric pressure is $1\times {10}^{5}N{m}^{-2}$. Find the heat supplied by the heater. The piston is massless and no friction exits between the piston and the cylinder. Neglect the heat loss through the lead wires of heater. The heat capacity of the heater coil is negligible.

• A. 120 J
• B. 600 J
• C. 720 J
• D. 300 J

Answer 1

The correct option is C 720 J

$\text{Solution}:$

The final pressure on the gas =atm pressure +pressure due to compression of spring

$P_2=P_{atm}+\frac{kx}{A}$

$⟹P_2={10}^5+\frac{8000\times 0.1}{8\times {10}^{-3}}=2\times {10}^{5}N{m}^2$

The final volume,

$V_2=V_1+xA=2.4\times {10}^{-3}+0.1\times 8\times {10}^{-3}=3.2\times {10}^{-3}{m}^{-3}$

Applying

$\frac{P_1{V}_1}{T_1}=\frac{P_2{V}_2}{T_2}$

$⟹T_2=\frac{P_2{V}_2{T}_1}{P_1{V}_1}$

$⟹T_2=\frac{2\times {10}^5\times 3.2\times {10}^{-3}\times 300}{{10}^5\times 2.4\times {10}^{-3}}=800K$.

Heat supplied by the heater is used for expansion of the gas, increasing its temperature and storing potential energy in the spring.

$\therefore$ Heat supplied =$P\Delta V+n{C}_v\Delta T+\frac{1}{2}k{x}^2={10}^5[0.8\times {10}^{-3}]+\frac{P_1{V}_1}{R{T}_1}{C}_v\Delta T+\frac{1}{2}k{x}^2$

$=80+\frac{{10}^5\times 2.4\times {10}^{-3}}{2\times 300}\times \frac{3}{2}\times 2\times 500+\frac{1}{2}\times 8000\times 0.1=720J$

$\text{Hence C is the correct option}$