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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 correct statementA. The density of gas cannot be uniform throughout the cylinderB. The density of gas cannot be uniform throughout the cylinder under isothermal conditionsC. The rate of change of density `|(drho)/(dh)|=(rhoMg)/(RT)`D. All of the above |
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Answer» Correct Answer - D 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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