question_answer

A gas in a sphere of radius 2r is transferred to another sphere of radius r. Find the pressure after transferring if initial pressure is P.

Answer:

               

Case - 1 (First sphere)	Case - 1 (First sphere)
\[{{r}_{1}}=2r\] \[{{P}_{1}}=P\]	\[{{r}_{2}}=r\] \[{{P}_{2}}=?\]
Temperature is constant

Temperature is constant \[\Rightarrow \] \[{{P}_{1}}{{V}_{1}}={{P}_{2}}{{V}_{2}}\] is applicable \[\Rightarrow \] \[{{P}_{1}}\times \frac{4}{3}\pi r_{1}^{3}={{P}_{2}}\times \frac{4}{3}\pi r_{2}^{3}\]                                 \[\left( \because {{V}_{sphere}}=\frac{4}{3}\pi {{r}^{3}} \right)\] \[\Rightarrow \] \[{{P}_{1}}\times r_{1}^{3}={{P}_{2}}\times r_{2}^{3}\] Substituting the values from data in the above equation, we get, \[\Rightarrow \] \[{{P}_{1}}\times {{(2r)}^{3}}={{P}_{2}}\times {{r}^{3}}\] \[\Rightarrow \]\[P\times 8{{r}^{3}}={{P}_{2}}\times {{r}^{3}}\] \[\Rightarrow \] \[{{P}_{2}}=8P\]