Entropy in Thermodynamics
In thermodynamics, entropy plays a crucial role in understanding the behavior of systems. Starting from Caratheodory's statement, this content delves into the concept of entropy, its expression as a function of other properties in reversible adiabatic processes, and the integration factor involved. Utilizing the ideal gas model, it generalizes to derive an expression for entropy, exploring conditions for a differential quantity to be path-independent and the significance of Caratheodory's new state variable.
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Presentation Transcript
Thermodynamics Ch7 : Entropy 7. Entropy from Caratheodory Statement Thermodynamics 1
Math. Intro: State property path dependent? P = 1 U is a state property: = dU dU U 1 U 2 1 A 2 1 2 1 2 Pd A B W depends on the path: Pd v v 2 2 A B B 1 dU is a total idfferential Pdv vis NOT a total differential In math. wording: v v ( ) ( )dy + , , f x y dx g x y Generally, the quantity: Is it a total differential? where :( x f ( ) ( ) ( ) ) ( ) If yes, we would write: = + , , , dh x y f x y dx g x y dy = = , ; , ; y h x g x y h y 2 Hence, Euler Condition : = = f y g x h x y ( ( ) ( ) ) = + Otherwise, we can find an integration factor A such as: , , dh A f x y dx g x y dy = = 2 = 1 f y g = x ydx+ ( x 2 The quantity: However: Is not a total diff. since: ) xdy Is a total diff. since: Example: 2 xdy + ydx 2 f y x g x ( ) As if we derived h = x2y giving dh = 2xydx + x2dy + Is path independent 2 x ydx xdy 12 Thermodynamics 2
A new state property v v=v v2v v=v v3 v v=v v1 P Every point on the P-v v chart represents ONE state P=P3 Lines of P = constant never intersect T=T3 P=P2 T=T2 Same for lines of v v = constant and T = constant P=P1 T=T1 v v Impossible Reversible Adiabatic Processes Possible Reversible Adiabatic Processes s=s3 Caratheodory: P Starting from a given state, We Cannot Construct 2 different processes that are reversible and adiabatic s=s2 We can define a styate property S (called Entropy) Such as: s=s1 v v d S = 0 (for a reversible adiabatic process) Thermodynamics 3
Entropy starting from Caratheodory How to express S As a function of other properties ? Reversible Adiabatic Processes P Special Case: Reversible Work of changing volume ONLY Perfect gas . W dt = P dV . s=s3 s=s2 Q dt = dU + P dV s=s1 v v . dS = Q dt = dU + P dV ? Can one put: NO! It is not a total differential Integration Factor: 1/T 1 T = ; 0 dS = (1/T) dU + (P/T) dV V U S is a state property, d S = 0(if reversible adiabatic) . P T mR V = = 0 U U V V dS = Qrev dt / T = (dU + PdV)/T Generalize: for all materials: Thermodynamics 4
Summary Entropy from Caratheodory Condition for a differential quantity to be Path independent Euler condition and integration factor Caratheodory New state variable Use ideal gas to derive integration factor Generalize to get an expression for Entropy T dS = dU + PdV Thermodynamics 5
