Question

An ideal gas is kept in a long cylindrical vessel fitted with a frictionless piston of cross-sectional area 10 cm² and weight 1 kg (figure). The vessel itself is kept in a big chamber containing air at atmospheric pressure 100 kPa. The length of the gas column is 20 cm. If the chamber is now completely evacuated by an exhaust pump, what will be the length of the gas column? Assume the temperature to remain constant throughout the process.

Answer 1

$\begin{array}{l}\\ \text{Atmospheric pressure}\mathrm{inside\; the\; cylinderical}\mathrm{vessel},P_0={10}^5\text{Pa}\\ {\text{A=10cm}}^2=10\times {10}^{-4}{\text{m}}^2\\ \text{Pressure due to the weight of the piston =}\frac{mg}{A}=\frac{1\times 9.8}{10\times {10}^{-4}}\\ P_1={10}^5+9.8\times {10}^3\\ V_1=0.2\times 10\times {10}^{-4}=2\times {10}^{-4}\\ \text{After evacution, external pressure above the piston =0}\\ P_2=0+9.8\times {10}^3\\ \text{Now,}\\ P_1{V}_1=P_2{V}_2\\ \text{Let L be the final length of the gas column. Then,}\\ V_2\text{=}10\times {10}^{-4}\text{L}\\ \Rightarrow ({10}^5+9.8\times {10}^3)\times 0.2\times 10\times {10}^{-4}=9.8\times {10}^3\times 10\times {10}^{-4}\text{L}\\ \text{⇒L=2}\text{.2 m}\end{array}$