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A very tall vertical cylinder is filled with a gas of molar mass M under isothermal conditions temperature T. the density and pressure of the gas at the base of the container is `rho_(0)` and `P_(0)`, respectively Select the incorrect statement if gravity is assumed to be constant throughout the containerA. Both pressure and density decreases exponetially with heightB. The variation of pressure is `P=P_(0)e^(-(Mgh)/(RT))`C. The variation of density `rho=rho_(0)e^((Mgh)/(RT))`D. The molecular density decreases as one moves upwards. |
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Answer» Correct Answer - C Considering the equilibrium of differential layer of a thickness dh and height h, we have `(P+dP)A+dmg=PA` or `A dP=-g dm=0grhoA dh` From ideal gas equation, we have `P=(rhoRT)/(M)` or `rho=(PM)/(RT)` From eqns. (1) and (2), we get `dp=-(PM)/(RT)g dh` or `In (P)/(P_(0))=-(Mg)/(RT)h` or `P=P_(0)e^(-Mgh//RT)` In a similar manner if we eliminate P with the of ideal gas equation, we get `dP=(drhoRT)/(M)` The eqn. becomes `drho(RT)/(M)=-rhog dh` or `int_(P_(0))^(rho) (drho)/(rho)=-(Mg)/(RT)int_(0)^(h)dh` or `rho=rho_(0)e^(-Mgh//RT)` |
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