Question

A vessel of volume V is evacuated by means of a piston air pump. One piston stroke captures the 1 volume $v_0$ . The pressure in the vessel is to be reduced to $\left(\frac{1}{n}\right)$ of its original pressure $P_0$ If the process n is assumed to be isothermal and air is considered an ideal gas, the number of strokes needed in the process is

• A. $\frac{In\eta }{ln\left(1-\frac{\Delta V}{V}\right)}$
• B. $\frac{ln\eta }{ln\left(1+V\right)}$
• C. $\frac{ln\eta }{ln\left(1+\frac{\Delta V}{V}\right)}$
• D. None of these

Answer 1

The correct option is C $\frac{ln\eta }{ln\left(1+\frac{\Delta V}{V}\right)}$

$\begin{array}{c}Let\rho bethedensityafterthefirststroke,theramainscons\tan t\\ V\rho =\left(V+\Delta V\right){\rho }_1\\ or,{\rho }_1=\frac{V\rho }{\left(V+\Delta V\right)}\\ similarly,if{\rho }_{2}isthedensityafter\sec ondstorke\\ V{\rho }_1=\left(V+\Delta V\right){\rho }_2\\ or,{\rho }_2=\left(\frac{V}{V+\Delta V}\right){\rho }_1={\left(\frac{V}{V+\Delta V}\right)}^2{\rho }_0\\ inthiswayafternthstroke,\\ {\rho }_n={\left(\frac{V}{V+\Delta V}\right)}^n{\rho }_0\\ \sin cepressure\alpha density,\\ {\rho }_n={\left(\frac{V}{V+\Delta V}\right)}^n{\rho }_0\\ \left(becuasetemperatureiscons\tan t.\right)\\ itisrequiredby\frac{{\rho }_n}{{\rho }_0}tobe\frac{1}{\eta }\\ so,\frac{1}{\eta }={\left(\frac{V}{V+\Delta V}\right)}^n\\ or,\eta ={\left(\frac{V+\Delta V}{V}\right)}^n\\ then,n=\frac{ln\eta }{ln\left(1+\frac{\Delta V}{V}\right)}\end{array}$

Hence,

option $\left(C\right)$ is correct answer.