Lecture 6: Entropy and 2 nd Law of Thermodynamics
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2 Lecture 6: Entropy and 2 nd Law of Thermodynamics Can we make heat work? 2 nd Law of Thermodynamics Entropy. Maximum entropy principle. Heat Engines. Carnot cycle. Computing Entropy
3 Summary of Processes. Process Type Constant Quantity Q W Isobaric p Isothermal T Isochoric V 0 Adiabatic pv γ 0 Can we find a better variable? Can we make heat work?
4 Stirling engine. A Efficiency = Work in cycle Absorbed Heat p D T 2 B T 1 C V 1 V 2 V Given amount of absorbed heat, work depends on shape (=efficiency) of cycle (absorbed heat) (released heat) Can we do better? 4
5 Limitations. Reversible/Irreversible Processes. 1. Glass of wine drops from the table. Pieces will never assemble back on their own. 2. Middle wall removed allowing gas to expand freely. Molecules will never never spontaneously return back to left part of container.
6 Limitations. Playing with 1 st Law of TD. 1. Direction of Heat Flow BUT! 1 st Law of TD does not forbid 2 nd process as long as energy is conserved. 2. Consider the World Ocean. It is warm after all! Take all its heat and convert into work. -- still OK with 1 st Law of TD Common sense tells us otherwise. Second Law of Thermodynamics 6
7 Second Law of Thermodynamics. Many formulations. Consider the following two. 1. Spontaneous (i.e. without external agent) transfer of heat from cold object to hot object is impossible. 2. No Perpetual Motion Machine of second type. (device that realizes full conversion of heat into useful work, without any side effect,, in contrast with real engines.) Let us formulate this mathematically. 7
8 Entropy. In 1 st Law of TD (for infinitesimal process) Q = not function of state,, depends on process There exist one more function of state = ENTROPY so that If T=const : Q=T S S S = 0 for adiabatic process S 0 for non-adiabatic Note: now have 4 variables (p,( V, T, S). S (any) two are still sufficient to describe state of system 8
9 Reversible and Irreversible Processes. Summary REVERSIBLE Series of quasi-static (equilibrium) steps No dissipation (e.g. no friction) Examples? Any process that can be represented as a path on a PV diagram; e.g Isochoric Isobaric Isothermal Adiabatic IRREVERSIBLE Series of complex, nonequilibrium steps Dissipation may be present - Examples? Any process that is sudden, spontaneous, uncontrolled, or involves dissipation Free expansion Explosion Heat transfer between a hot and a cold object Volume changes of gas with a piston that has friction 9
10 Calculating Entropy Changes. Reversible Processes Adiabatic process? General process: (T( i, V i, p i ) --> > (T( f, V f, p f ) What is entropy change for: Isothermal process? Isochoric process?
11 Principle of Maximum Entropy. Intuitively: Entropy = Measure of Disorder With time, entropy of isolated system: increases in irreversible process, remains the same in reversible process Disorder grows! If system is NOT at thermal equilibrium, but consists of equilibrium subsystems with S i, introduce S Σ = Σ i S i In isolated system (fixed internal energy),, state of thermal equilibrium corresponds to absolute maximum of total entropy, i.e. S=(S Σ ) max If you reach thermal equilibrium with untidy roommate, that would correspond to maximum possible disorder. 11
12 How can we extract heat? Heat Engines. Want engine to work repeatedly Want cyclic process! Heat engine takes some substance (e.g. gas) through cyclic process during which: Heat Q H absorbed from hot reservoir Work W + done by engine Heat Q C dumped into cold reservoir Negative work W - done by engine to bring gas to original state. In a cyclic process, it is impossible to convert all the energy extracted from an equilibrium system into work. Some part of this energy must be transferred to a colder system.
13 Internal combustion engine. Typical cycle consists of 4 processes a) Burning of fuel (isothermal expansion) b) Fast expansion (adiabatic) c) Compression to original pressure (isothermal process) d) Compression to original temperature (adiabatic process) 13
14 Carnot heat engine takes gas through reversible cyclic process consisting of 4 steps: Isothermal expansion at T=T H Adiabatic expansion from T=T H to T=T C Isothermal compression at T=T C Adiabatic compression Carnot Heat Engine from T=T C to T=T H T K = 1 C < 1 T H
15 Carnot heat engine and 2 nd Law of TD. A corollary to the second law of thermodynamics: NO REAL ENGINE CAN EXCEED THE EFFICIENCY OF A CARNOT ENGINE OPERATING BETWEEN THE SAME HEAT RESERVOIRS Q: can an engine (real or ideal) ever have an efficiency of 1 (I.e. 100 %)? Why/why not?
16 Calculating Entropy Changes. Irreversible Processes An irreversible process is characterized by a series of non-equilibrium states -- thermodynamic parameters such as T,p are ill-defined So: how to calculate S? Entropy is a state function -- so S S only depends on the initial and final equilibrium states -- NOT THE PATH! Strategy: choose a reversible path connecting the initial and final states and determine S. 16
17 p Sample problem. Free expansions. 2 moles of a gas in a thermally insulated box freely expands, tripling its volume in the process. What is the change in entropy? p 1, V 1, T 1 p 2, V 2, T 1 V Q = 0; W = 0; so, U = 0 Final T = initial T Choose a reversible, isothermal path S = = = 3V V 0 0 dq T nrt T nrt T 3V V 0 3V V = 2R ln 3 = 0 = nr ln 3 = 0 0 3V V 0 0 dv V dv V pdv T 18.3J/K
18 Two identical 500 g copper blocks A and B of mass are initially at 20 0 C and 80 0 C. They are placed in thermal contact and allowed to come to thermal equilibrium. What is the change in entropy? (Specific heat capacity of copper = 400 J/kg. K) 323 mcdt S A = 293 T = mcln 323 S B = mcln = J/K = (0.5kg)(400J/kg.K)(0.098) =19.6 J/K Sample problem. Heat transfer. What reversible process can we use? S total = ( )J/K = +1.8 J/K
19 What we learned Reversible/Irreversible Processes Entropy 2 nd Law S 0 S = nc v T ln T f i + V nr ln V f i Heat Engines (Carnot) K = W Q H K = 1 T T C H
20 The law that entropy always increases, holds, I think, the supreme position among the Laws of Nature. If someone points out to you that your pet theory of the Universe is in disagreement with Maxwell s Equations then so much the worse for Maxwell's equations. If it is found to be contradicted by observation well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation. --Sir Arthur Stanley Eddington, The Nature of the Physical World (1927)
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