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LECTURE 10 ENTROPY AND THE SECOND LAW OF THERMODYNAMICS Text Sections 21.2, 21.3 Sample Problems 21.1, 21.2 Suggested Questions 3, 4 Suggested Problems 29P Summary Entropy of an equilibrium state Quasistatic (reversible) processes Changes in entropy during processes Entropy as a state function The Second Law of Thermodynamics Specific objectives • • • Explain the difference between quasistatic (reversible) and non-quasistatic (irreversible) processes Calculate the change in entropy for quasistatic (reversible) processes and for nonquasistatic (irreversible) processes using the entropy change for quasistatic processes Explain the Second Law of Thermodynamics and its relation to entropy LECTURE 10 ENTROPY AND THE SECOND LAW OF THERMODYNAMICS Entropy of an equilibrium state In the last lecture we defined entropy in terms of the probability of various microscopic configurations, i.e. in terms of the microscopic disorder or complexity of the system. Historically entropy was first introduced in (macroscopic) thermodynamics in order to describe the behaviour of steam engines. We will follow this macroscopic approach in the last three lectures. The Zeroth Law indicates that it is possible to associate a thermodynamic parameter, temperature, with each equilibrium state. The First Law indicates that it is possible to associate another thermodynamic parameter, internal energy, with each equilibrium state. The Law is a statement of conservation of energy. As we have seen there are processes which are consistent with the First Law, i.e. which conserve energy, but which do not occur naturally. The Second Law and entropy are required to determine which processes can occur. Entropy, like temperature and energy, is associated with equilibrium states. Unlike energy however it is not conserved in real processes occurring in isolated (closed) systems when an internal constraint is removed. As we saw last lecture, for example, entropy increases in a fee expansion of a gas. The process is macroscopically irreversible. *** Changes in entropy during processes In Lecture 5 we discussed the fact that real processes of gas systems cannot be plotted on a p – V diagram because all states other than the initial and final states are not equilibrium states. Only these two states can be plotted (see Fig 21-2). If the process is carried out very slowly in a large number of small steps then the system is never far from equilibrium and the process can be approximated by a line on a p – V diagram joining the end points. The idealised process represented by the line is called a quasistatic process (i.e. almost static). (Some textbooks including HRW call it a reversible process. This can lead to confusion as reversible is also used with a different meaning.) Quasistatic processes are reversible in the sense that the line can be traversed in either direction. Examples of non-quasistatic (irrreversible) processes include: free expansion of a gas (see Fig 21-1); and two systems with different initial temperatures coming to thermal equilibrium. Similar considerations to the above apply to other types of thermodynamic system described by thermodynamic parameters analogous to p and V. Some examples of such systems were given in Lecture 5. The change in entropy of a system in a quasistatic process between an initial ( i ) and a final ( f ) equilibrium state is defined by f ∆ S = Sf − Si = ∫ i dQ T where dQ is the infinitesimal amount of heat transferred to the system and T is the temperature (in kelvin!) at that stage of the process. If we know the entropy at any one equilibrium state then this equation can be used to find the entropy at all other equilibrium states of the system. Even if we don’t know the entropy at any equilibrium state we can find the entropy at all equilibrium states except for an additive constant (c.f. potential energy). Once this is done the change in entropy during any process (whether it is quasistatic or not) between two equilibrium states can be found. *** Entropy as a state function Guided by the insight provided by the microscopic description we have asserted that entropy is a property of an equilibrium state. In a purely macroscopic description this has to be determined by careful experiments on many different systems (c.f. internal energy). It is possible however to verify that entropy is a state function in the special case of an ideal gas. Consider a quasistatic process from an initial equilibrium state (i ) to a final equilibrium state (f ). The First Law applies to each of the infinitesimal sub-processes, i.e. d Eint = dQ − dW . Hence dQ d E int dW = + T T T n cV d T pd V dT dV = + = n cV + nR T T T V by using the Ideal Gas Law. Therefore for the full process ∆ S = S − Si f = ∫ i dQ = n cV T Tf ∫ Ti dT + nR T Vf ∫ Vi dV V . This type of integral was evaluated in Lecture 6 (see also derivation of Eqn 20-10 of HRW). Therefore T V S f − Si = n cV l n f + n R l n f Ti Vi . The right-hand side of the equation is independent of the thermodynamic parameters for the intermediate states of the process. The difference in entropy therefore depends only on the initial and final equilibrium states and does not depend on the particular quasistatic process between them. For example, the entropy change is the same for the two processes shown in Figs 21-1 (non-quasistatic) and 21-3 (quasistatic). The equation can also be used to find the change in entropy during a non-quasistatic process between the two equilibrium states. If the initial and final temperatures are equal, i.e. Tf = Ti , then the first term in the above equation is zero and V ∆ S = Sf − Si = n R ln f Vi . This includes the special case where the process is isothermal, i.e. T = Tf = Ti . For a free expansion process in which the final volume is twice the initial volume ∆ S = n R ln ( 2 ) in agreement with the microscopic result. If the initial and final volumes are equal, i.e. Vf = Vi , then the second term in the above equation is zero and T ∆ S = Sf − Si = n cV ln f Ti . This includes the special case where the process is isochoric, i.e. V = Vf = Vi . The Second Law of Thermodynamics This Law can be stated in many ways. For example: The total entropy of all systems taking part in a process never decreases. It remains the same only if the process is quasistatic. The entropy of an isolated (closed) system can never decrease. It remains the same only if all internal processes are quasistatic. Note that real processes are never exactly quasistatic. Total entropy (often grandly called the entropy of the Universe even though only a few parts of the Universe are involved) always increases if any change occurs. The first statement of the Law above emphasises that it is necessary to consider the changes in entropy of all systems taking part in the process. The entropy of one or more systems can decrease but only if the entropy of one or more other systems increase sufficiently for the total entropy to either remain the same or increase. For example, in the quasistatic isothermal expansion of an ideal gas system, a positive amount of heat Q is transferred to the gas at a constant temperature T . During the process the entropy of the gas increases by Q / T . The heat is extracted from a heat reservoir at temperature T . This is a quasistatic isothermal process for that system also. The entropy of the heat reservoir decreases by Q / T . The total entropy of the two systems (gas and reservoir) does not change. If the gas system is compressed, its entropy decreases and that of the reservoir increases by the same amount. Alternatively the gas system and the reservoir can be regarded as together forming an isolated system. Since all internal processes are quasistatic the entropy of that isolated system does not change. Entropy is sometimes called the “Arrow of Time” since it indicates whether a process or its time-reversed form will occur in nature. Only one of these two real processes can occur spontaneously.