Thermodynamic Processes: The Limits of Possible

Thermodynamics put severe restrictions on what processes involving a change of the thermodynamic state of a system (U,V,N,…) are possible.

In a quasi-static process system moves from one equilibrium state to another via a series of other equilibrium states .

All quasi-static processes fall into two main classes: reversible and irreversible processes .

Processes with decreasing total entropy of a thermodynamic system and its environment are prohibited by Postulate II

Notes Graphic representation of a single thermodynamic system

Phase space of extensive coordinates The fundamental relation S(1) =S(U (1) , X (1) ) of a thermodynamic system defines a hypersurface in the coordinate space S(1) S(1)

U(1) U(1)

X(1) X(1) S(1) – entropy of system 1 (1) (1) (1) (1) (1) U – energy of system 1 X = V , N 1 , …N m – coordinates of system 1 Notes Graphic representation of a composite thermodynamic system

Phase space of extensive coordinates The fundamental relation of a composite thermodynamic system S = S (1) (U (1 ), X (1) ) + S (2) (U-U(1) ,X (2) ) (system 1 and system 2). defines a hyper-surface in the coordinate space of the composite system S(1+2) S(1+2) U (1,2) X = V, N 1, …N m – coordinates U of subsystems (1 and 2)

X(1,2)

(1,2) S – entropy of a composite system X U – energy of a composite system Notes Irreversible and reversible processes

If we change constraints on some of the composite system coordinates,

(e.g. Xi and Xj) the composite system relaxes to a new equilibrium state with the maximum possible entropy consistent with the new constraints

Reversible process, (1+2) Irreversible process, S(1+2) S entropy increases Entropy is constant (most processes are (idealized situation, (1,2) like that) B X (1,2) there is no real Xi i drive for the system as Postulate II Is not at work) B

(1,2) X (1,2) Xj A j A

Notes Heat and Entropy

If we have flow of heat between two subsystems, then: δQ1 = −δQ2

δQ1 = T1dS 1 δQ2 = T2dS 2

If temperatures of the two subsystems are equal to each other: T1 = T2

dS 1 = −dS 2 dS = dS 1 + dS 2 = 0

then the process is reversible.

Heat flow between systems with different temperatures is always irreversible .

Notes Heat capacity and heat flow

Let us consider a situation of T1>T 2 before we remove adiabatic wall between the two systems.

After new equilibrium is reached, the final temperature of the two systems is Tf.

Let us also make a simplifying assumption that heat capacity (CV = dU/dT ) at constant volume for these systems does not depend on temperature:

∆U1 = C1(Tf −T1 ) ∆U 2 = C2 (Tf −T2 )

Notes Equilibrium temperature

∆U1 = C1(Tf −T1 ) ∆U 2 = C2 (Tf −T2 )

Total energy of the composite system must be conserved:

∆U = ∆U1 + ∆U 2 = (C1 + C2 )Tf − (C1T1 + C2T2 )= 0

(C1T1 + C2T2 ) Tf = ()C1 + C1

Notes Entropy Change

δQ C dT  T  1 1 1 ln ln ln  f  dS 1 = = ∆S1 = C1()()Tf − ()T1 = C1   T1 T1  T1 

For the composite system:  T   T   f   f  ∆S = ∆S1 + ∆S2 = C1 ln   + C2 ln    T1   T2   +   +   C1T1 C2T2   C1T1 C2T2  ∆S = C1 ln   + C2 ln   > 0  T1()C1 + C2   T2 ()C1 + C2 

Notes Heat flow is irreversible

 +   +   C1T1 C2T2   C1T1 C2T2  ∆S = C1 ln   + C2 ln   > 0  T1()C1 + C2   T2 ()C1 + C2 

Taking for simplicity, C1=C 2 :

 +   +   ( + )2   T1 T2   T1 T2   T1 T2  ∆S = C ln   + C ln   = C ln   > 0  2T1   2T2   4T2T1 

Irreversible process T1 > T2

Notes Utilizing Entropy to do Work

Let us start thinking about how to extract work from a thermodynamic system.

We have a system that goes form one equilibrium state to another: A→B.

As it undergoes this transformation, it does work -δW and releases heat -δQ.

We would like to know, what is the maximum work that this system can do in the process A→B.

In particular, we would like to compare reversible and irreversible processes .

Notes Heat Engine Model

We model the environment of the system as two objects: - reversible work source (sink) RWS - reversible heat source (sink) RHS Heat engine model

∆U = U B –UA -δQ RHS -δW System A →B RWS Heat sink Work sink TRHS T

Rigid walls, so no work can Adiabatic walls, so no heat can be be done on the heat sink ! transferred to the work sink! ( dS RWS =0 ) Notes Energy-efficient use of Entropy

Problem: to minimize energy E spent to lift a weight using a stretched rubber band. Lift by hand, no help Lift by hand at constant Lift by warming the from the rubber band T, rubber band helps rubber band

E = W '= ? EΓ=const = Q = ? E0 = W = Mgh T =const

Spontaneous heat flow

h h h

min (EΓ=const , ET =const )= ? Notes