Question

The figure shows a cylindrical container containing oxygen (γ = 1.4) and closed by a 50-kg frictionless piston. The area of cross-section is 100 cm², atmospheric pressure is 100 kPa and g is 10 m s⁻². The cylinder is slowly heated for some time. Find the amount of heat supplied to the gas if the piston moves out through a distance of 20 cm.

Answer 1

Given:
Mass of the piston (m) = 50 kg
Adiabatic constant of the gas, γ = 1.4
Area of cross-section of the piston (A) = 100 cm²
Atmospheric pressure (P₀) = 100 kPa
g = 10 m/s²
Distance moved by the piston , x = 20 cm
Work done by the gas,
$$\begin{gathered} d\mathrm{W}=Pdv \\ \text{The p}\mathrm{ressure}\left(p\right)\mathrm{is}\mathrm{because}\mathrm{of}\mathrm{two}\mathrm{factors}:\text{the}\mathrm{first}\mathrm{is}\mathrm{the}\mathrm{initial}\mathrm{pressure}\mathrm{and}\text{the}\mathrm{second}\mathrm{is}\mathrm{the}\mathrm{weight}\mathrm{of}\mathrm{the}\mathrm{piston}. \\ \mathrm{Therefore}, \\ W=\left(\frac{mg}{\mathrm{A}}+P_0\right)\times Adx \\ W=\left(\frac{50\times 10}{100\times {10}^{-4}}+{10}^5\right)\times 100\times {10}^{-4}\times 20\times {10}^{-4} \end{gathered}$$
W = (5 × 10⁴ + 10⁵) × 20 × 10⁻⁴
W = 1.5 × 10⁵ × 20 × 10⁻⁴
W = 300 J

Hence, nRdT = P$∆V$ = 300
$$\begin{gathered} \Rightarrow d\mathrm{T}=\frac{300}{nR} \\ \mathrm{So},d\mathrm{Q}=n{\mathrm{C}}_{p}dT=n{c}_p\times \left(\frac{300}{nR}\right) \\ \text{U}\mathrm{sing}{C}_p-C_v=Rand\frac{C_p}{C_v}=\gamma , \\ d\mathrm{Q}=\frac{n\gamma R300}{(\gamma -1)n\mathrm{R}} \\ d\mathrm{Q}=\left(\frac{300\times 1.4}{0.4}\right)=1050\mathrm{J} \end{gathered}$$