| A vessel of volume \[V\] is evacuated by means of a piston air pump. One piston stroke captures the 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 is assumed to be isothermal and air is considered an ideal gas the number of strokes needed in the process is |
A) \[\left[ \frac{\ell nn}{\ell n\left( 1-\frac{{{v}_{0}}}{V} \right)} \right]\]
B) \[\left[ \frac{\ell nn}{\ell n\left( 1+\frac{{{v}_{0}}}{V} \right)} \right]\]
C) \[\left[ \frac{\ell n\left( 1-\frac{{{v}_{0}}}{V} \right)}{\ell n} \right]\]
D) none of these
Correct Answer: B
Solution :
| \[VP-(V+{{v}_{0}}){{P}_{1}}\]\[\Rightarrow \]\[{{P}_{1}}=\left[ \frac{VP}{V+{{v}_{0}}} \right]\] |
| and \[V{{P}_{1}}=\left( V+{{v}_{0}} \right){{P}_{2}}\]\[\Rightarrow \] \[{{P}_{2}}={{\left[ \frac{V}{V+{{v}_{0}}} \right]}^{2}}P\] |
| Therefore \[{{\left[ \frac{V}{V+{{v}_{0}}} \right]}^{n}}P=\frac{P}{n}\] |
| After simplifying, we get |
| \[n=\left[ \frac{\ell n\,\,n}{\ell n\left( 1+\frac{{{v}_{0}}}{V} \right)} \right]\] |
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