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.

Go Back to Class 11 Maths Home page Go Back to Class 11 Physics Home page

link to us