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State second law of thermodynamics . Describe the working of carnot engine . Obtain an expression for the efficiency . |
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Answer» Solution :Second Law of Thermodynamics : First law of thermodyamics is based on Law of conservation of ENERGY . While second law of thermodynamics gives information about the transformation of heat energy . So , there are TWO conventional statements of second law depending on common experience . 1) Kelvin - Plank statement : It is impossible for an engine working in a cyclic process to extract heat from a hot body and to convert it completely into work . 2) Clausius Statement : It is impossible for a self ACTING machine , unaided by any external agency to transfer heat from a cold body to a hot reservoir . In other words heat cannot by itself flow from a colder body to a hotter body . Carnot's Engine : Carnot's engine works on the principle of reversible process with -in the TEMPERATURES`T_(1)andT_(2)` It consists of four continuous processes . The total process is known as Carnot Cycle . Step 1 : In Carnot cycle the 1st step consists of isothermal expansion of gases. So temperature T is constant , P , V changes areshown as `P_(1)V_(1)T_(1)rarr^("to")P_(2)V_(2)T_(1)` work done in isothermal process `W_(1rarr2)=Q_(1)=muRT_(1)log_(e)(V_(2)/(V_(1)))"" .....(1)` Step : 2 In this stage gases will expand adiabatically . So energy to the system Q is constant . So , P,V relation is `P_(2)V_(2)T_(1)overset("to")rarrP_(3)V_(3)T_(2)` Work done in adiabaticallyprocess `W_(2rarr3)=(muR(T_(1)-T_(2)))/(gamma-1)"" ...(2)`  In this stage gases will be compressed isothermallt . So`P_(1)V` changes are `P_(3)V_(3)T_(2)overset"to"rarrP_(4)V_(4)T_(2)` work done in isothermal compression `W_(3rarr4)=Q_(2)=muRTlog_(e)((V_(3))/(V_(4)))""...(3)` Step 3 : In this stage gases will be compressed isothermally , So`P_(1)V` changes are `P_(3)V_(3)T_(2)overset("to")rarrP_(4)V_(4)T_(2)` Work done in isothermal compression `W_(3rarr4)=Q_(1)-muRTlog_(e)((V_(3))/(V_(4)))""....(3)` Step 4 : In the fourth stage the gas suffers adiabatic compression and returns to original stage . So , P,V changes are`P_(4)V_(4)T_(2)overset("to")rarrP_(1)V_(1)T_(1)` `W_(4rarr1)=(muR(T_(2)-T_(1)))/(gamma-1)""...(4)` Total work done in Carnot Cycle `W=W_(1,2)+W_(2,3)+W_(3,4)+W_(4,1)` Total work done `W=muRT_(1)log((V_(2))/(V_(1)))` `+(muR)/(gamma-1)(T_(1)-T_(2))-muRT_(2)log(V_(3)/(V_(4)))-(muR)/(gamma-1)(T_(1)-T_(2))` `therefore W=muRT_(1)log((V_(2))/(V_(1)))-muRT_(2)log(V_(3)/(V_(4)))` The total work done `W=Q_(1)-Q_(2)` i.e., the difference to heat energy absorbed from source and heat energy given to sink Efficiency of Carnot engine `eta=("work done by Carnot engine")/("heat energy supplied")` `thereforeeta=(Q_(1)-Q_(2))/(Q_(1))" or " eta=1-(Q_(2))/(Q_(1))=1-(T_(2))/(T_(1))` |
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