Second Law of Thermodynamics

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1 Dr. Alain Brizard College Physics II (PY 211) Second Law of Thermodynamics Textbook Reference: Chapter 20 sections 1-4. Second Law of Thermodynamics (Clausius) Heat flows naturally from a hot object to a cold object; heat will not flow spontaneously from a cold object to a hot object. Heat Engine: any device that changes heat into mechanical work. Heat Pump: any device that allows heat to flow from a cold object to a hot object. Reversible Cycles A cycle Γ is defined as a sequence of processes that trace a closed curve on the PV plane (see Figure below), i.e., the system begins and ends in the same thermodynamic state. A cycle evolves either in the clockwise direction or in the counter-clockwise direction; a cycle is reversible if it traces the same closed curve independently of the cycle s evolution. 1

2 Since a cycle begins and ends in the same thermodynamic state (P 0,V 0,n 0,T 0 ), the change of internal energy for a (reversible) cycle is zero and, thus, the First Law of Thermodynamics implies that U Γ 0 W Γ Q Γ, i.e., the work done by a system as it evolves along the cycle Γ is equal to the heat absorbed by the system during the cycle Γ. Note that W Γ positive if cycle evolves in clockwise direction negative if cycle evolves in counter-clockwise direction Second Law of Thermodynamics (Kelvin-Planck) No device is possible whose sole purpose is to transform a given amount of heat completely into work. The Second Law of Thermodynamics implies that the heat Q Γ absorbed during a cycle must be expressed as Q Γ Q out, where is the heat input and Q out is the heat output during the cycle Γ. If work is done by the system during the cycle (i.e., W Γ > 0), we find that >Q out and represents heat absorbed by the system at a high temperature and Q out represents heat released by the system at a low temperature. The ideal efficiency ε Γ of the cycle Γ is defined as ε Γ W Γ 1 Q out. The Second Law of Thermodynamics, therefore, states that the ideal efficiency of any cycle describing a heat engine can never be 100 %. A heat pump, in contrast, involves a system absorbing heat from a cold object (at a low temperature ), while work W Γ < 0 is being done on the system, and releasing heat Q out + W Γ to a hot object (at a high temperature ). The ideal efficiency of a heat pump (i.e., a refrigerator) is defined in terms of the coefficient of performance (CP) CP Γ W Γ Q out. 2

3 The Figure below shows the difference between a heat engine and a heat pump. Carnot Cycle Sadi Carnot ( ) discovered an idealized reversible cycle with the highest ideal efficiency. The diagram below shows the four-step Carnot cycle (C : a b c d a). The first step a b represents an isothermal expension from (,P a )to(,p b )at constant temperature, where P a nr P b. The second step b c represents an adiabatic expansion from (,P b ) on the 3

4 isothermal to (V c,p c ) on the isothermal, where P b P c ( ) 5 The third step c d represents an isothermal compression from (V c,p c )to(,p d ) at constant temperature, where P c V c nr P d. The fourth step d a represents an adiabatic compression from (,P d ) on the isothermal to (,P a ) on the isothermal, where P a P d ( ) 5 The two adiabatic processes yield the following identity V c Step a b: Isothermal expansion U ab 0 and W ab nr ln ( ) Vb Q ab > 0 Step b c: Adiabatic expansion Q bc 0 and U bc 3 2 nr ( ) W bc < 0 Step c d: Isothermal compression U cd 0 and W cd nr ln ( ) Vc Q cd < 0 Step d a: Adiabatic compression Q da 0 and E da 3 2 nr ( ) W da > 0 For the Carnot cycle C : a b c d a, wefind [ ( ) Vb U C 0 and W C nr ln ln ( )] Vc Q C 4

5 We can now calculate the ideal efficiency of the Carnot cycle ε C W C 1 Q out 1 ln(v c / ) ln( / ) If we now use the identity shown above, we find ln(v c / )ln( / ) and thus Carnot s Theorem 0 < ε C 1 < 1 (since 0 < < ). All reversible engines operating between the same two constant temperatures and < have the same efficiency. Any irreversible engine operating between the same two fixed temperatures will have an efficiency less than this. Otto Cycle The Otto cycle involves a four-step sequence involving an adiabatic expansion (a b), an isovolumetric process (b c), an adiabatic compression (c d), and an isovolumetric process (d a); here, V c and while T a ( / ) 2/3 T b and T d ( / ) 2/3 T c. The efficiency of the Otto cycle is given as ε Otto 1 Q out 1 Va 3 1 T b T a 1 T c T d < ε C 1 T c T a 5

Chapter 12. The Laws of Thermodynamics

Chapter 12. The Laws of Thermodynamics Chapter 12 The Laws of Thermodynamics First Law of Thermodynamics The First Law of Thermodynamics tells us that the internal energy of a system can be increased by Adding energy to the system Doing work