12 The Laws of Thermodynamics

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1 June 14, The Laws of Thermodynamics Using Thermal Energy to do Work Understanding the laws of thermodynamics allows us to use thermal energy in a practical way. The first law of thermodynamics is a statement about the conservation of energy and the second law is a statement about entropy, which usually increases and can only decrease when energy is expended by an outside system. An understanding of this law lets us determine the practical limits of heat engines and refrigerators Heat and Internal Energy Thermal energy is that portion of internal energy that changes with temperature. Also when two systems at different temperatures are put into thermal contact, thermal energy will transfer as heat from the higher temperature to the lower temperature until both systems are at the same temperature. We then say equilibrium has been achieved. 109

2 Work and Heat It is important to realize that energy can be transferred between two systems even when no heat is involved. For example, two sticks rubbed together create additional thermal energy in each stick. Also a transformation of one form of energy to another can increase the thermal energy of the system. Nuclear and chemical forms of energy can be converted into thermal energy. Also frictional forces can put thermal energy into the system. It is an energy transfer due to work Work and Heat A macroscopic ideal gas that is internally in equilibrium can be described with the equation of state PV =nrt. These variables can be related to mechanical situations if we notice that work W = F x = (F/A)A x = P V, so W = P V (12. 1) If the pressure of a system remains constant, then the amount of work it does is just P(V f - V i ) = P V. Thus if V = 0 then W= 0. When P varies the work equals the area under the curve of a P versus V diagram. Also the work done depends upon the path between the initial and final states. (We have already discussed how friction and processes involving heat are not conservative as far as mechanical energy is concerned.) Both heat and work are processes for transferring energy so neither are conserved in thermodynamic processes. When there is no thermal energy transfer to a system, we call the process an adiabatic processes. When there 110 The Laws of Thermodynamics

3 The First Law of Thermodynamics is not change in pressure, it is called an isobaric process The First Law of Thermodynamics W out U f Q in If we denote the internal energy of the system with the letter U, the heat with Q, and the work done by the system with W, then U = Q - W, which is a statement about the conservation of energy of the system and is known as the first law of thermodynamics. Thus we have U = U f U i = Q W (12. 2) U i Note: It is the change in U that counts and we don t need to know the details of just how the internal energy changed. For isolated systems, U remains constant. If there are cyclic processes so we come back to the initial state, even though the system is not isolated, the change in U is zero. So Q = W Heat Engines and The Second Law of Thermodynamics A heat engine is a device that converts thermal energy to other useful forms such as electrical or mechanical energy. In a heat engine a substance goes through a cycle of absorbing and rejecting heat and doing work. The work is equal to the difference in the heat in from a hot reservoir and heat out into a cold reservoir, i.e. the net heat, The Laws of Thermodynamics 111

4 Reversible and Irreversible Processes W = Q h Q c (12. 3) Work done is the area enclosed by the curve in a P,V diagram. The thermal efficiency is defined as the ratio of the net work done to the heat absorbed during one cycle, i.e. e W Q h Q c = = = 1 Q h Q h Q c Q h (12. 4) One way of stating the second law is: It is impossible to construct a heat engine that, operating in a cycle, produces no other effect than the absorption of heat from a reservoir and the performance of an equal amount of work Reversible and Irreversible Processes Broken egg Irreversible process A reversible process is one that leaves the system and the environment unchanged in going through a cycle. Otherwise the process is irreversible. All natural processes are known to be irreversible. If the changes occur very slowly so the environment and system are nearly always in equilibrium, the process is nearly reversible. There can be no dissipative effects present in a true reversible process, so none actually exist 12.6 The Carnot Engine Sadi Carnot in 1824 using the theory of thermodynamics described a cycle for a heat engine that has the highest efficiency possible. No other engine can get 112 The Laws of Thermodynamics

5 Entropy more work out of an input of heat than the Carnot engine. The cycle is bounded by four reversible processes, two of which are adiabatic (no heat lost or gained) and two are isothermal (no change in temperature). He showed the efficiency of the cycle to be T h T c e c = = 1 T h T c T h (12. 5) In this equation T must be in absolute temperature units, e.g. kelvins. The higher the temperature of the input heat and the lower the temperature of the output heat, the higher the efficiency will be. Real engines have frictional and heat losses so they are never quite as efficient as a theoretical Carnot engine Entropy Temperature is a state function associated with the zeroth law of thermodynamics. Energy and its conservation is another state function and is associated with the first law. Another state function is entropy and it is associated with the second law. Entropy has to do with the amount of order or disorder of a system. The Clausius definition of change in entropy is S = Q r T (12. 6) where the subscript r refers to reversible processes. One of the main outcomes of the concept of entropy is that real processes are not reversible and they increase entropy. The system tends to more disorder and entropy is a measure of that disorder. The change of The Laws of Thermodynamics 113

6 Entropy and Disorder entropy for a reversible cycle, such as a Carnot engine cycle, is zero. But all natural processes increase entropy and the entropy of the universe is constantly increasing Entropy and Disorder Entropy is a state function and, therefore, is dependent only upon the state of a system and not how the system got there. Thus, while most processes are irreversible, in calculating the change in entropy between two states, we can devise a way of calculating over infinitesimal differences that are reversible, so we can use the definition we have, namely S = Q r /T. Thus the change in entropy can be determined from the change of state of the system regardless of how that change occurred. Ordered States Disordered States Isolated systems tend toward disorder, and entropy is a measure of that disorder. Disordered states in nature are much more probable than ordered states because there are many more of them, that is, the number of ways we can have disorder is much higher than the number of ways we can have order. Thus disorder is more probable, so entropy measures the degree to which a system has progressed towards the most probabilistic state. In light of this view of entropy it is found that entropy can be related to the probability, W, of a system being in a particular state so that S = k B lnw (12. 7) 114 The Laws of Thermodynamics

7 Concept Statements and Questions where k is Boltzmann s constant (k B = J/K) We are able to conclude from this that entropy is a measure of microscopic disorder. The second law of thermodynamics is really a statement of what is most probable, not of what must be Concept Statements and Questions 1. Internal energy of a system can also include potential energy of various forms as well as kinetic energy, and such energy is considered to be thermal energy. 2. Are there some situations when the first law of thermodynamics can be broken? Give an example if it is possible. 3. How do pressure, P, and volume, V, relate to work done on a gas or by a gas? 4. What is meant by an irreversible process? How could it be made reversible? Why are all processes in nature irreversible according to the definition? 5. It is impossible to make a heat engine more efficient than the Carnot engine. What effect does temperature have on the efficiency of an engine? Which law or laws of thermodynamics rule out perpetual motion machines? 6. Thermal energy will not naturally flow from a cold object to one with a higher temperature. The Laws of Thermodynamics 115

8 Hints for Solving the Problems 7. Is the coefficient of performance, COP, the same thing as efficiency? If not, how does it differ? 8. Entropy always increases in the universe. If it decreases in a system, it is because energy or work is put into that system which causes an even greater increase in entropy elsewhere Hints for Solving the Problems General Hints 1. Consider there are many ways in which the second law of thermodynamics can se stated. Think about them. They are all expressing the same ideas. 2. The Carnot engine is not 100% efficient, but it is the most efficient engine that can be made to operate between two temperature reservoirs. 3. Study the examples carefully to help prepare you for working the problems on your own. 4. Remember that energy has to be conserved whether there is heat added or work done on a system. Hints for Solving Selected Problems 1-9. The area under a P,V diagram gives the work. Also you may need the relationship of PV = nrt for an ideal gas. P V gives work but V P doesn t If work is done on a gas the internal energy of the gas increases. If the gas does work, the internal energy decreases. Study examples 12.2 and The Laws of Thermodynamics

9 Hints for Solving the Problems Use the equations developed for the efficiency of the ideal heat engine (Carnot engine). This sets the limit for performance of all other engines. You can work with either the heat transferred or the absolute temperatures to get the efficiencies. There is a formula for each Calculate the heat transferred for the two states of the system, and divide by T to get entropy change. 42. The loss in KE could be considered the heat transferred Once you have figured the probability according to the problem, the entropy is obtainable from S = k B ln W. The Laws of Thermodynamics 117

10 Hints for Solving the Problems 118 The Laws of Thermodynamics

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