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Lect 6
- 1. Entropy Two reversible adiabatic can not intersect each other CA B C isotherm RA AB and CB are two reversible adiabatic and they intersect at B. AC is a reversible isotherm. Through AC we can transfer heat to the cycle ABC which is A close cyclic process. From the cycle ABC there is no heat loss but work out put. So This violates 2nd law of thermodynamics. Hence two RA can not intersect each other. p v i f a b RA Rev isotherm Any rev path Through i, f there is a Rev process. We construct two rev adiabatic through i and f, then a rev isotherm ab such that: Area under iabf = area under if, First law for if : , πππ = π’ π β π’π + π€ππ Process iabf: πππππ = π’ π β π’π + π€ππππ π€ππ = π€ππππ, π π πππ = πππππ p v πππππ = πππ + π ππ + π ππ = π ππ =πππ Thus any rev path can be substituted by a reversible Zig-zag path, between the same end states, consisting of a rev adiabatic + rev isotherm+ another rev adiabatic.
- 2. Clausius theorem p v Any rev cyclic process RA Rev isotherm T1 T2 Lets us consider a smooth close curve which represents a rev cycle. Then we take two Elemental parts from this cycle as shown in the picture (upper part and lower part). The upper part cab represented by an isotherm+ two RA and similarly the lower part of the curve. The upper isotherm is at a temp say T1 and the lower isotherm is at T2. Now this elementary process can be thought of as a Carnot engine which receives heat dq1 at T1 and rejects heat dq2 at T2. So for this elementary cycle ππ1 π1 = ππ2 π2 ππ ππ π€π πππ π€ππ‘β π‘βπ Signs of heat transfer then we get ππ1 π1 + ππ2 π2 = 0. If we add all the elementary cycles such that we cover the entire cyclic process then we get : dq1 dq2 ππ1 π1 + dq2 T2 + dq3 T3 + β― . dqn Tn = 0 or ππ π = 0 , hence ππ π ππ π πππππππ‘π¦ ππππ π’, π, π£ πππ β π€βππβ is called entropy , ds. R1 R2 a b π π ππ π π 1 + π π ππ π π 2 = 0, ο π π ππ π π 1 = π π ππ π π 2 Hence entropy is a property, independent of path So, ππ = ππ πππ£ π or, π π ππ π = π π ππ = π π βπ π Clausius theorem
- 3. ππ = ππ πππ£ π ; If ππ πππ£ = 0, then the process is called rev adiabatic and then ds=0 Hence s=Const; T S A B CD ππ ππ2 RA AB is a general process Either rev or irreversible, All other processes are rev. Consider an elemental cycle, as shown in the picture. ππ = heat supplied at π1, ππ2 = βπππ‘ ππππππ‘ππ ππ‘ π2 1 β ππ2 ππ πππ¦ β€ 1 β ππ2 ππ πππ£ ππ2 ππ πππ¦ β₯ ππ2 ππ πππ£ ππ πππ¦ ππ2 β€ ππ πππ£ ππ2 ππ πππ¦ ππ2 β€ π π2 ; π ππππ ππ πππ£ ππ2 = π π2 Or, Or, Or, Or, ππ πππ¦ π β€ ππ2 π2 ο ππ πππ¦ π β€ ππ , πππ πππ¦ ππππππ π For a reversible process: ππ π = ππ For any other process : ππ β₯ ππ π For any cyclic process: ππ π β€ ππ ππ π β€ 0 πΌπ ππ π = 0, cycle is reversible, ππ π < 0, cycle is irreversible and feasible ππ π > 0, cycle is impossible Clausius Inequality
- 4. Entropy principle For any process undergone by the system, ππ β₯ ππ π , if the process is reversible then ππ = ππ π For isolated system, dq=0, so, ππ β₯ 0, if isolated process is reversible, π‘βππ ππ = 0, π = π For irreversible process, ππ > 0, π π πππ‘ππππ¦ πππ€ππ¦π πππππππ ππ Causes of entropy increase: for a closed system entropy increases due to (a) External interaction, (b) internal irreversibility ππ = ππ ππ₯π‘ + ππ πππ‘ = ππ π + ππ πππ‘; so, ππ > ππ π ; ππππ¦ πππ πππ£πππ ππππ ππππππ π ππ πππ‘ = 0 ππ = ππ π + ππ πππο π 2 β π 1 = 1 2 ππ π + π πππ1β2 ππ ππ‘ = π ππ£ π + π πππ Entropy change Entropy transfer Entropy generation dq T π πππ π 2 β π 1 For an open system ππ π π πππ π π ππ π π + π ππ£ π + π πππ = π π π π + ππ ππ£ ππ‘ , for steady state ππ ππ£ ππ‘ = 0 ππ π π + π ππ£ π + π πππ = π π π π + π2 π 2 β π1 π 1 Integral form of the equation Change of entropy within the CV system
- 5. π» π π relation First law and second law combined: ππ = ππ’ + πππ£ ο πππ = ππ’ + πππ£ ππ = ππ’ π + π π ππ£ ππ = ππ’ π + π π£ ππ£ ππ πππ£ = πππ π 2 β π 1 = π π£ ππ π + π π£ ππ£ π 2 β π 1 = π π£ ln π2 π1 + π ln π£2 π£1 1st Tds relation Only for ideal gases β = π’ + ππ£ πβ = ππ’ + πππ£ + π£ππ πβ = πππ + π£ππ Tππ = πβ β π£ππ 2nd Tds relation ππ = πβ π β π£ π ππ π 2 β π 1 = π π ππ π β π π ππ ο π π ln π2 π1 β π ln π2 π1 Only for ideal gases