Question

Consider a piston cylinder arrangement with spring at its natural length. Due to heat transfer, the gas expands until the piston hits the stop. The mass of the piston is 10 $kg$ and area is 78 $c{m}^2$. Initial and final specific internal energies are 214 $kJ/kg$ and 337 $kJ/kg$. Then, find the correct options.($K=10000N/m;g=9.8m/s^2;x=5cm$)

• A. Initial pressure of gas is 112.5 $kPa$.
• B. Work done by gas on the piston is approximately 56.5 $J$.
• C. Work done by gas on the piston is 100 $J$.
• D. Heat transfer for 0.5 $gm$ gas is 118 $J$.

Answer 1

Answer: (A), (B), (D)

Heat transfer for 0.5 $gm$ gas is 118 $J$.

Initially,
$P_{gas}=P_0+\frac{mg}{A}={10}^5+\frac{10\times 9.8}{78\times {10}^{-4}}={10}^5+1.25\times {10}^4=(100+12.5)kPa=112.5kPa$
Work done by piston
$W={\int }_0^x(P_{0}A+mg+kx)dx={\int }_0^{5\times {10}^{-2}}({10}^5\times 78\times {10}^{-4}+10\times 9.8+{10}^{4}x)dx={\int }_0^{5\times {10}^{-2}}(780+98+{10}^{4}x)dx=780(5\times {10}^{-2})+98(5\times {10}^{-2})+\frac{{10}^4\times {(5\times {10}^{-2})}^2}{2}=39+4.9+12.5=56.4\approx 56.5JQ=m\Delta U+w\Delta U=337-214=123kJ/kg\therefore Q=0.5\times {10}^{-3}\times 123\times {10}^3+56.5=61.5+56.5=118J$