Entropy and the 2 nd Law of Thermodynamics

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1 Entropy and the 2 nd Law of Thermodynamics Heat Engines A heat engine is a system that converts thermal energy into work. It includes a working substance (e.g., a gas) that goes through a cycle. Q in =Q h > 0 Q out =Q c < 0 W = Q in +Q out =Q 'in Q out =Q h Q c Since the work is done by the engine, W > 0 and Q h > Q c Efficiency The efficiency of a heat engine is the ratio of the positive net work (done during the cycle) to the heat absorbed. Since ΔU = 0 it follows Q NET = W NET with Q NET = Q in + Q out = Q in - Q out. The efficiency of the heat engine is given by e = W NET or e= Q in Q out =1 Q out Q in Q in Q in

2 Carnot cycle A Carnot engine is a hypothetical engine that operates in a reversible cycle of an ideal gas between hot and cold reservoirs. A Carnot cycle consists of two isotherm and two adiabatic processes. The individual processes are: A B: Isothermal expansion. Heat Q h is absorbed from the high temperature reservoir at T h. B C: Adiabatic expansion. Temperature drops from T h to T c. C D: Isothermal compression. Heat Q c is released to the low temperature reservoir at T c. D A: Adiabatic compression. Temperature increases from T c back to T h. It can be shown that for the Carnot cycle Q c / Q h = T c / T h. Thus, the efficiency is e = 1 T c T h The efficiency of the Carnot heat engine is found to be the maximum possible efficiency of any heat engine that absorbs heat at T hot and dumps heat at T cold. All other heat engines (Diesel, Otto, Stirling, etc...) working using ideal gases have a lower efficiency than the Carnot cycle. A Carnot engine absorbs heat at 500 o C and dumps heat at 25 o C. What is the maximum efficiency? e=1 T c = =0.386 T h If the engine takes in heat at the rate of 10 kw, what is the power output?

3 W =e Q in Power= W Δt =e Q in =(0.368 )(10 kw )=3.68 kw Δt The Otto cycle The Otto cycle is closely related to a car engine. A stroke is movement of the piston corresponding to a transformation from a minimum to a maximum volume, or vice versa. Since it take four strokes of the piston to complete the cycle: (0 to 1), (1 to 2), (3 to 4), (1 to 0), the Otto engine is a four stroke engine. The piston absorbs heat from a high temperature reservoir, converts some of the heat into work, and dumps heat into a low temperature reservoir. A figure above displays an ideal Otto cycle. A more realistic Otto cycle is shown in the figure below.

4 Second Law of Thermodynamics There are several statements for the second law of thermodynamic, some of which are equivalent, and others are more general. Consider the following cases 1- Two objects at the same temperature are in thermal contact. After some time, it does not happen that one become colder and the other hotter. 2- Start with a mixture of salt and pepper in a glass. Mix them for some time, it does not happen the separation of salt and pepper such that the top part of the glass contains pepper and the bottom part contains salt. 3- A chandelier falls from the ceiling and all the light bulbs break into pieces. It doesn t naturally occur the inverse of this process where all the pieces get back together and the chandelier moves up into the ceiling. 4- Air is made of moving molecule. It doesn t naturally occur that all the sudden all the air molecules move in one corner of the room and you remain breathless. 5- Perpetual motions (of several kinds). None of these cases conflicts with conservation of energy. In other words they could all happen without conflicts with conservation of energy. So why they do not actually happen in nature? The second law of thermodynamics is the answer: it imposes a constrain on the conservation of energy. 2 nd Law - Clausius statement: It s not possible for heat to transfer from a cooler object to a warmer object. This statement explains case number one. Reversible and Irreversible Processes A process of a system from the initial state A to the final state B is reversible if it satisfies the following three conditions: 1 It is possible for the process to occur in the reverse order B to A. 2 The intermediate steps from B to A are the same of the steps from A to B. The changes in temperature between steps are infinitesimal. 3 Once the process from A back to A is completed not only the system but also the environment returns to the same initial state.

5 s of a reversible process is the friction-less motion of solids (as a pendulum oscillating without friction) or the slow isothermal expansion (or compression) of gases. A process which does not satisfy all the three conditions above is irreversible. In reality, kinetic friction is always present: for example if a gas expanses its temperature changes, making all natural processes to be is irreversible. As such we have 2 nd Law Irreversibly statement: All processes in nature are irreversible. This statement explains case number three and five. However, a process can be approximated as reversible if it can be done in small, well-defined steps. In the case of the isothermal expansion of a gas, if the cylinder is in thermal contact with a large thermal reservoir during the expansion and if the expansion is done very slowly, then the gas temperature is nearly the same as the reservoir. Even if reversible processes are not realistic, they are useful idealizations and tools, similar to the ideas of point-like particles or elastic collisions. initial: A thermally insulated container is separated in two parts by a wall: one is filled with the molecules of an ideal gas, the other is vacuum. final: A hole is poked through the wall allowing the gas molecules to move and occupy the entire container. The gas expansion is free since no work has been done by the gas during the expansion. The inverse process can occur in nature and it is relatively simple to achieved: by removing heat from the system the gas goes back to the same initial pressure and volume. The processes is irreversible since the steps backwards are not necessarily the same; furthermore the initial and final state of the environment are not the same because of the heat absorbed by the environment.

6 Heat Engine A heat engine cannot convert the Q in from at hot source completely to work. Some heat Q out must be released at a lower temperature. 2 nd Law - Kelvin-Planck statement: Q out cannot be zero. Since Q out 0 no heat engine can have an efficiency of 100% (e = 1). The reason is because of kinetic friction is always present and friction generates heat. Since each transformation is irreversible (friction is present for each transformation), more heat is actually lost by the heat engine and the real efficiency is less than the ideal engine s efficiency: e real engine < e ideal engine The opposite process instead is possible: 100% of work can be converted into heat. This statement explains case number five. Refrigerator and Heat Pump A refrigerator and a heat pump are heat engines operating in reverse. Q out =Q h < 0 Q in =Q c > 0 W = Q in +Q out =Q in Q out =Q c Q h Since the work is done on the system (by electricity), W < 0 and Q c < Q h.

7 Calculate the work done by a refrigerator to absorb 20J of heat and release 70J. By choosing the correct signs: Q c =20 J, Q h = 70 J W = 20 70= 50 J. For both cases (refrigerator and the heat pump), heat is transferred from the cold reservoir to the hot reservoir. As the process occurs, the temperature of the hot reservoir increases while the temperature of the cold reservoir decreases. The refrigerator focuses on removing heat from the cold reservoir: it extracts the heat Q c from the box with the food, and dumps the heat Q h into your kitchen. Similarly a AC unit extracts heat from your house and dump it on the outside. The hose is present in order to carry the heat Q h outside. The coefficient of performance (COP) is the ratio of what you want (Q c or Q h ) divided by what you pay for, i.e. the work done by electricity which by convention is defined to be positive W E = - W. For a refrigerator we have COP(refrigerator ) = Q Q c = c. W E Q h Q c The heat pump focuses on adding heat to the hot reservoir: it extracts the heat Q c from outside your house and dump heat Q h into your house. For a heat pump we have COP(heat pump ) = Q h W E = Q h Q h Q c. The Clausius statement of the 2 nd Law implies it is impossible to have a system which transfers heat from a cold to a hot reservoir without doing work on the system.

8 Entropy The entropy S is an intrinsic quantity of a system like the mass or volume or color. It is a state function as its value depends only of the actual state of the system and not on how the system got to that state, i.e. is path independent. Entropy is an extensive quantity as the total entropy of a multiple systems is simply the summation of the individual entropies. Reversible process In thermodynamic, not the entropy itself (see later) but only the change of entropy of a system can be defined. If a system exchanges (either absorbs or releases) an amount of heat Q during a reversible process at a fixed temperature, then the change of entropy of the system is Δ S = Δ Q T units = J/K or cal/k If heat is absorbed, then ΔS > 0. If heat is lost, then ΔS < g of ice melts at 0 o C. What is the change in entropy of the water? Δ S = Q T = ml f T = (50 g)(80 cal/ g) 273 K = 14.7 cal /K =61.3 J / K If the temperature changes during a reversible process (reminder: reversible means infinitesimal temperature differences between steps), then the change of entropy is obtain as the sum of the changes of entropy of each step at different temperatures. A substance of given mass m and specify heat c absorbs heat as its temperature increases from T i to T f. Find the expression for the change of entropy of the substance. The heat absorbed by the substance is ΔQ = mcδt, and since the temperature is not constant, calculus must be used to calculate the change in entropy. Δ S = mc ln( T f T i )

9 Irreversible process What is the change of entropy for an irreversible process? Since entropy is a state function the change of entropy is independent of how the process occurs (path independent). A process takes a system from the initial state 1 to the final state 2. The process can happen in various ways (paths). Path A indicates a reversible process, Path B indicates an irreversible process. We have Δ S A = Δ S B So for a given irreversible process is possible to calculate its change of entropy by defining a reversible process with same initial and final states and then using the equation of the reversible process. 100 cal of heat is transferred from a reservoir at 100 o C to a reservoir at 0 o C. Assuming that the reservoirs are large enough so that their temperatures stay constant, what is the change in entropy of the total system given by the two reservoirs? The system is irreversible since heat only flows in the direction T hot to T cold. ΔS=ΔS hot +ΔS cold = Q hot + Q cold = 100cal T hot T cold 373 K +100cal 273 K = cal/ K 100 g of water at 20 o C is mixed with 100 g of water at 80 o C. What is the net change in entropy? Although this is an irreversible process, we could get the same final result (200 g of water at 50 o C) by slowly warming the cool water from 20 o C to 50 o C and slowly cooling the warm water from 80 o C to 50 o C. Thus, ΔS = ΔS hot +ΔS cold = =(100 g )(1 cal/ g o C )ln ( ) +(100 g)(1 cal/ go C )ln ( ) =9.75 cal / o C 8.88 cal/ o C=0.87 cal / o C

10 Note that in both previous examples ΔS > 0, if heat flowed from the cold to the hot, then ΔS would be negative. This cannot occur spontaneously. 2 nd Law Entropy statement: The total entropy of a isolated* system increases in all natural processes Δ S 0 Where ΔS = 0 corresponds to a reversible ideal process. *several resources refer to an isolated system as closed system, be careful. For any given system, the total system = (the system + its environment) is an isolated system often referred as the universe. For example your coffee cup is the system, while your room and house and the Earth and our galaxy and everything else out there forms the environment. Considering altogether the total system is the universe. The term universe is used since the actual entire universe is a isolated system because no heat can enter or leave it and no work can either be done on or done by the universe. For a cycle process CP the change in entropy is zero S CP = 0. This is true for both reversible and irreversible processes. But there is a difference which has to do with the change of entropy of the corresponding environment. If a cycle process takes place the total change of entropy of the isolate total system is: Δ S = Δ S CP +Δ S ENV If the cycle process is reversible then ΔS ENV = 0 and ΔS = (ΔS CP + ΔS ENV ) = 0. If the cycle process is irreversible then ΔS ENV > 0 and ΔS = (ΔS CP + ΔS ENV ) > 0. Recall the example of free expansion of a gas, if heat is dumped into the environment the gas returns to its initial conditions of pressured volume (cyclic process). Since the process is irreversible, the entropy of the universe increases. From the previous equation we see how the total entropy of the universe can stays constant only if cyclic reversible processes were to occur in nature, which is not the case. It is possible for a process (reversible or irreversible) to occur such that the entropy of the system decreases. For example when using a refrigerator heat is removed from the fridge, resulting in entropy to decrease inside the fridge. But in order for this process to occur work must be done from the outside (the environment) by the use of electrical energy resulting in the increase of entropy of the environment.

11 The second law states that even if it is possible for the entropy of a system (the fridge) to locally decrease, the total entropy of the system (fridge) + the environment always increases. Heat death of the Universe From the second law follows that the entropy of the universe progressively increases toward a maximum hypothetical value. When such values will be reached all systems contained in the universe come into thermal equilibrium at a uniform temperature. After that point, no further changes involving the conversion of heat into useful work are be possible. Entropy and information A more fundamental and general definition of entropy is found in statistical mechanics. In this case entropy is related to the amount of disorder of an isolated system: the higher the disorder, the higher the entropy. An intrinsic value of the amount of entropy of a system is defined as S = k B ln W The value of W indicates the number of possibilities for a macroscopic configuration M to occur. The higher is W the most likely that configuration occurs. Note how in the context of thermodynamic it is only possible to calculate the change of entropy ΔS of a thermodynamic system and not the intrinsic value S of it. Tossing a coin four times, the possible outcomes (the macroscopic configuration M) are: M H: head, T: tail W 4 heads HHHH 1 3 heads, 1 tail HHHT, HHTH, HTHH, THHH 4 2 heads, 2 tails HHTT, HTHT, HTTH, THHT, THTH, TTHH 6 1 heads, 3 tail HTTT, HHTH, HTHH, THHH 4 4 tails TTTT 1 For the case M = two heads and two tails W = 6 which is the highest value for W. That means W = 6 is the configuration most favorable to occur when tossing a coin four times. It also corresponds to the highest entropy.

12 2 nd Law Disorder statement: Processes in nature occur in such a way that the disorder of the system naturally increases, such systems spontaneously evolve towards thermodynamic equilibrium, the configuration with maximum entropy. Put inside a container one cup of salt, then add one cup of pepper, shake the container. The macroscopic configuration most favorable to occur is the mixture between salt and pepper, you can keep shaking the container but it s very unlikely to obtain the initial configuration again. For the initial macroscopic configuration the value of W is very small, while the value of W for the mixture is large since it can be realized in many different ways. Third Law of Thermodynamics The third law of thermodynamics is concerned with the limiting behavior of systems as the temperature approaches absolute zero. 3 rd Law of Thermodynamics: It is impossible by any procedure, no matter how idealized, to reduce the temperature of any system to zero degree Kelvin, in a finite number of finite operations. While a temperature of absolute zero does not exist in nature and we cannot achieve it in the laboratory, the concept of absolute zero is critical for calculations involving temperature and entropy. As of this writing, the record-low temperature was achieved 1999 by the YKI-group of the Low Temperature Laboratory at Aalto University in Finland. They cooled a piece of rhodium metal to about K