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A heating element of piston resistance r is fitted inside an adiabatic cylinder which carries a frictionless piston of mass m and crosssection A as shown in diagram. The cylinder contains one mole of an ideal diatomic gas. The current flows through the element such that the temperature rises with time t as DeltaT =alpah t+1/2beta t^2 ? (alphaand betaare constants), while pressure remains constant. The atmospheric pressure above the piston is P_0. Then |
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Answer» the RATE of increase in internal energy is `5/2 R ( alpha + betat )` ` therefore` Rateof increase in INTERNALENERGY ` (dU)/( d t)= n C_V (d T)/(d t)` here ` n=1, dT=alpha t+1/2betat^2therefore(dT)/( d t)= alpha+ betat` fordiatomicgas ` C_V=5/2R ,C_P= 7/2R ` `therefore(dU )/( d t)= 1 xx5/2R xx( alpha+ betat)= 5/2R ( alpha+ betat )` heatsuppliedby theheatingelement to thegas atconstantpressure ` d Q= n C_(P )d T ` Rateof hetasupplied ` ( dQ)/(d t)= nC_P( d t)/( d t) = 1 xx (7)/(2)R xx ( alpha+ beta t )` ( using (i )) ` =7/2R ( alpha+ beta t )` LetI becurrentflowingin theelement ` thereforeI^2r =7/2R( alpha+ betat )` ` I^2 = 7/( 2 r)R (alpha + betay)impliesI= sqrt( 7/(2 r)R (alpha+ betat))` accordingto idealgas equation`PV = nRT ` At contain pressure`PdVn RT ` ` PdV= nR( alphat + 1/2beta t^2 )` workdone` w =pdV ` ` =n R( alpha t + 1/2beta t^2 )` LetF beforceexertedbythegasonthe pistonof areaA ` thereforeP = ( F)/( A)orF = pA ` letthepistonmovedupwardsby adistancex . ` thereforeW = F xorx =W /F= (nR)/(pA )( alphat +1/2beta t ^2)`( using (ii ) ) veloity` v=(dx )/ (dt)=(nR)/( PA)( alpha+ betat )` Acceleration` a = ( dv )/( dt ) = ( nR) /(PA )beta ` |
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