8. Work done in Isothermal process

  • In an isothermal process temperature remains constant.
  • Consider pressure and volume of ideal gas changes from (P1, V1) to (P2, V2) then, from first law of thermodynamics
         ΔW = PΔV
    Now taking ΔV aproaching zero i.e. ΔV𔾴 and suming ΔW over entire process we get total work done by gas so we have
         W = ∫PdV
    where limits of integration goes from V1 to V2
    as PV = nRT we have P = nRT / V
         W = ∫(nRT/V)dV
    where limits of integration goes from V1 to V2
    on integrating we get,
         W=nRT ln(V2/V1)               (3)
    Where n is number of moles in sample of gas taken.

9. Work done in an Adiabatic process

  • For an adiabatic process of ideal gas equation we have
         PVγ = K (Constant)               (14)
    Where γ is the ratio of specific heat (ordinary or molar) at constant pressure and at constant voluume
         γ = Cp/Cv
  • Suppose in an adiabatic process pressure and volume of a sample of gas changs from (P1, V1) to (P2, V2) then we have
    Thus, P = K/Vγ
  • Work done by gas in this process is
         W = ∫PdV
    where limits of integration goes from V1 to V2
    Putting for P=K/Vγ, and integrating we get,
         W = (P1V1-P2V2)/(γ-1)          (16)
  • In and adiabatic process if W>0 i.e., work is done by the gas then T2< T1
  • If work is done on the gas (W<0) then T2 > T1 i.e., temperature of gas rises.

10. Heat Engine and efficiency

  • Any device which convents heat continously into mechenical work is called a heat engine.
  • For any heat engine there are three essential requirements.
    (i) SOURCE : A hot body at fixed temperature T1 from which heat engine can draw heat
    (ii) Sink : A cold body, at a fixed lower temprature T2, to which any amount of heat can be rijectd.
    (iii) WOEKING SUBTANCE : The material, which on being supplied with heat will do mechanical work.
  • In heat engine, working substances, could be gas in cylinder with a moving piston.
  • In heat engine working substance takes heat from the sorce, convents a part of it into mechanical work, gives out rest to the sink and returns to the initial state. This series of operations constitutes a cycle.
  • This cycle is represented in fig below

    Heat Engine and efficiency
  • Work from heat engine can be continously obtained by performing same cycle again and again.
  • Consider,
         Q1 - heat absorbed by working substance from sorce
         Q2 - heat rejected to the since
         W - net amount of work done by working substance
         Q1-Q2 - net amount of heat absorbed by working substance.
         ΔU = 0 since in the cycle Working Substance returns to its initial condition.
    So on application of first law of thermodynamics
         Q1-Q2 = W
  • Thermal efficiency of heat engine
         η= work output in energy units / Heat input in same energy units
          = W / Q1 = (Q1-Q2 )/ Q1
    Or, η = 1-(Q2/Q1)                    (17)
    from this equation it is clear that
         Q = 1 for Q2=0
    and there would be 100% conversion of heat absorbed into work but such ideal engines are not possible in practice.

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