question_answer

Two identical containers each of volume \[f(x)=\sqrt{\sin x},g(x)={{x}^{2}}\] are joined by a small pipe. The containers contain identical gases at temperature \[f(x)=|x|,g(x)=\sin x\] and pressure \[f(x)=\sqrt{x},g(x)={{\sin }^{2}}x\]. One container is heated to temperature \[f(x)=\sin \sqrt{x},g(x)={{x}^{2}}\]while maintaining the other at the same temperature. The common pressure of the gas is P and n is the number of moles of gas in container at temperature \[f:R\to R\]

A) \[f(x)=\frac{x}{\sqrt{1+{{x}^{2}}}},\]                      

B) \[\frac{x}{\sqrt{1+{{x}^{2}}}}\]

C) \[\frac{x}{\sqrt{1+3{{x}^{2}}}}\]               

D) \[f(x)=\frac{{{a}^{x}}+{{a}^{-x}}}{2}\]

Correct Answer: B , C

Solution :

Initially for container A, \[{{V}_{0}}\] For container B, \[{{T}_{0}}\] \[{{P}_{0}}\]\[2{{T}_{0}}\] Total number of moles \[2{{T}_{0}}\] Since, even on heating the total number of moles is conserved Hence,   \[p=2{{p}_{0}}\]                                                                                                              ?(i) If p be the common pressure, then For container A, \[p=\frac{4}{3}{{p}_{0}}\]\[n=\frac{2}{3}\frac{{{p}_{0}}{{V}_{0}}}{R{{T}_{0}}}\]\[n=\frac{3}{2}\frac{{{p}_{0}}{{V}_{0}}}{R{{T}_{0}}}\] For container A,\[\frac{pV}{RT}\] \[2:\pi \]\[\pi :2\] Substituting the value of \[\lambda \] and \[\lambda /2\]in Eq. (i) we get\[\lambda /3\] Number of moles in container A (at temperature \[\lambda /4\]) \[\lambda /9\] \[{{10}^{-10}}W/{{m}^{2}}.\]

Solution :

Initially for container A, \[{{V}_{0}}\] For container B, \[{{T}_{0}}\] \[{{P}_{0}}\]\[2{{T}_{0}}\] Total number of moles \[2{{T}_{0}}\] Since, even on heating the total number of moles is conserved Hence,   \[p=2{{p}_{0}}\]                                                                                                              ?(i) If p be the common pressure, then For container A, \[p=\frac{4}{3}{{p}_{0}}\]\[n=\frac{2}{3}\frac{{{p}_{0}}{{V}_{0}}}{R{{T}_{0}}}\]\[n=\frac{3}{2}\frac{{{p}_{0}}{{V}_{0}}}{R{{T}_{0}}}\] For container A,\[\frac{pV}{RT}\] \[2:\pi \]\[\pi :2\] Substituting the value of \[\lambda \] and \[\lambda /2\]in Eq. (i) we get\[\lambda /3\] Number of moles in container A (at temperature \[\lambda /4\]) \[\lambda /9\] \[{{10}^{-10}}W/{{m}^{2}}.\]