WEBVTT 00:00:03.520 --> 00:00:08.460 You have probably heard entropy defined or described as "disorder." The usual example 00:00:08.460 --> 00:00:13.280 is that of a college dorm room, which, without regular tidying, becomes "messier" or "less 00:00:13.280 --> 00:00:18.100 ordered" over time. Supposedly the entropy of the messy room is higher than that of the 00:00:18.100 --> 00:00:24.340 tidy room. This analogy is easy to picture, but it's misleading. In this video, you'll 00:00:24.340 --> 00:00:27.380 learn a more accurate description of entropy and understand how it relates to the concept 00:00:27.380 --> 00:00:27.550 of spontaneity. 00:00:27.550 --> 00:00:31.880 This video is part of the Governing Rules video series. A small number of rules describe 00:00:31.880 --> 00:00:36.040 the physical and chemical interactions that are possible in our universe. 00:00:36.040 --> 00:00:41.390 Hi. My name is John Lienhard and I am a professor in the Department of Mechanical Engineering 00:00:41.390 --> 00:00:42.660 at MIT. 00:00:42.660 --> 00:00:49.660 Today, I'm going to talk to you about entropy, a fascinating, but often confusing topic. 00:00:51.180 --> 00:00:55.520 In order to understand the topic of this video, you should be familiar with the idea that 00:00:55.520 --> 00:01:02.320 energy is quantized and the thermodynamic definition of a system and its surroundings. 00:01:02.320 --> 00:01:06.950 After watching this video, you should be able to describe, at a basic level, the concept 00:01:06.950 --> 00:01:11.899 of a microstate. You should also be able to discuss what entropy measures in a conceptual 00:01:11.899 --> 00:01:13.240 way. 00:01:13.240 --> 00:01:20.240 First, what do we mean by a spontaneous process? In thermodynamics, a spontaneous process is 00:01:20.880 --> 00:01:25.810 one that will occur without any outside intervention given enough time. 00:01:25.810 --> 00:01:31.279 In the world around us, many everyday events proceed in a particular manner. We would call 00:01:31.279 --> 00:01:37.020 them spontaneous. You have observed spontaneous processes yourself, but because they seem 00:01:37.020 --> 00:01:41.450 so natural, you may not have taken particular note of them. 00:01:41.450 --> 00:01:45.630 For example, think of an inflated balloon that hasn't been tied and is simply pinched 00:01:45.630 --> 00:01:52.630 between someone's fingers. Once the person releases the balloon, what is going to happen? 00:01:53.729 --> 00:01:58.840 Experience tells us that the gas inside the balloon will rapidly escape from the opening, 00:01:58.840 --> 00:02:03.639 moving from high pressure to low pressure. This will propel the balloon through the air, 00:02:03.639 --> 00:02:08.910 until finally, we are left with a deflated balloon. The gas that was once in the balloon 00:02:08.910 --> 00:02:12.970 is now dispersed throughout the surroundings. 00:02:12.970 --> 00:02:18.800 You have probably also seen food coloring or hydrophilic dye dropped into water. What 00:02:18.800 --> 00:02:25.390 happens? From experience, you may know that the dye disperses. 00:02:25.390 --> 00:02:30.730 You may also have had some experience removing hot pans from the stovetop. While they come 00:02:30.730 --> 00:02:37.020 off of the stovetop hot, we know they will eventually cool. Here, we see a liquid crystal 00:02:37.020 --> 00:02:43.640 in the pan change color, first as the pan is heated, and then again, as the pan cools. 00:02:43.640 --> 00:02:47.829 Experience tells us in which direction these everyday events will proceed. But what about 00:02:47.829 --> 00:02:51.470 processes with which we don't have experience? 00:02:51.470 --> 00:02:56.120 For example, it would be nice if we had a way of knowing whether or not a given chemical 00:02:56.120 --> 00:02:59.940 reaction will happen at given conditions. 00:02:59.940 --> 00:03:04.300 The 2nd law of thermodynamics can help us with this. 00:03:04.300 --> 00:03:09.900 The 2nd law of thermodynamics states that during any spontaneous process, the total 00:03:09.900 --> 00:03:15.990 entropy change of a system and its surroundings is positive. In other words, the entropy of 00:03:15.990 --> 00:03:22.180 the "universe," that is, the system plus surroundings, can only increase. 00:03:22.180 --> 00:03:27.290 But what is entropy? Is entropy a magical force that overturns your furniture and creates 00:03:27.290 --> 00:03:30.600 havoc in your office or home? No. 00:03:30.600 --> 00:03:34.020 Entropy is a measure of the number of possible ways energy can be distributed in a system 00:03:34.020 --> 00:03:40.840 of molecules. Molecules in a system at equilibrium have the same average energy. However, at 00:03:40.840 --> 00:03:46.069 a given instant in time, it is highly unlikely that all of the molecules have the same exact 00:03:46.069 --> 00:03:48.350 energy. 00:03:48.350 --> 00:03:53.319 Molecules in a system are constantly interacting and transferring energy amongst each other. 00:03:53.319 --> 00:03:58.360 As a result, one molecule may have a certain amount of energy at one instant and at the 00:03:58.360 --> 00:04:03.640 next; it could have more or less. Depending on the energy the molecule has, it will be 00:04:03.640 --> 00:04:08.150 able to access different energy levels. 00:04:08.150 --> 00:04:11.900 The total energy of the system, determines what energy levels will be accessible to the 00:04:11.900 --> 00:04:17.289 molecules. Higher energy levels will not be accessible because the energy required to 00:04:17.289 --> 00:04:21.189 reach them is not available. 00:04:21.189 --> 00:04:25.099 So when we say that entropy is a measure of the number of possible ways energy can be 00:04:25.099 --> 00:04:31.249 distributed in a system of molecules, we have to account for all of the possible combinations. 00:04:31.249 --> 00:04:35.580 And the way we do that is by considering the microstates available to the molecules in 00:04:35.580 --> 00:04:40.460 the system. Let's use an analogy to understand the term 00:04:40.460 --> 00:04:41.680 "microstate". 00:04:41.680 --> 00:04:48.069 Let's say that you have two dice. What are all of the possible sums for a pair of dice? 00:04:48.069 --> 00:04:55.069 Pause the video here and take a moment to jot them down. 00:04:55.439 --> 00:05:02.080 Okay, you should have a list that looks something like this. We would call these sums possible 00:05:02.080 --> 00:05:08.199 macrostates of our system -- the macrostate doesn't tell us what each individual die reads 00:05:08.199 --> 00:05:13.569 when we roll them, just the total, or "macroscopic view" if you will. 00:05:13.569 --> 00:05:18.300 What are all of the possible dice combinations that will produce each of those sums? For 00:05:18.300 --> 00:05:23.240 example, we can produce the sum of three by rolling a one on the first die and a two on 00:05:23.240 --> 00:05:28.900 the 2nd die. Or, we can roll a 2 on the first die and a one on the 2nd die. So there are 00:05:28.900 --> 00:05:32.558 two combinations that will produce the sum of 3. 00:05:32.558 --> 00:05:37.219 The dice combinations that produce the remaining sums are shown here. 00:05:37.219 --> 00:05:43.759 We would call each of these combinations "microstates" that correspond to each macrostate. The microstate 00:05:43.759 --> 00:05:47.669 gives us information about the individual conditions of each die. 00:05:47.669 --> 00:05:52.669 We see that the most likely macrostate, a sum of 7, has the greatest number of possible 00:05:52.669 --> 00:05:53.139 microstates. 00:05:53.139 --> 00:05:57.849 Do you think that the entropy change for the system (the cold bar) was positive, negative, 00:05:57.849 --> 00:05:58.969 or equal to zero? 00:05:58.969 --> 00:06:05.189 Please pause the video here and discuss your reasoning with a classmate. 00:06:05.189 --> 00:06:12.189 Let's start with the system first. The transfer of energy to the cold bar will allow the molecules 00:06:43.339 --> 00:06:50.339 in the cold bar to access new energy levels that they could not reach before, increasing 00:08:34.229 --> 00:08:40.309 the number of possible microstates for that system. So we would suspect that the entropy 00:08:40.309 --> 00:08:43.179 change for the system is positive. 00:08:43.179 --> 00:08:45.620 But what about the surroundings? 00:08:45.620 --> 00:08:52.140 The total entropy of a system and its surroundings has to increase if the process is spontaneous. 00:08:52.140 --> 00:08:57.200 Let's use a very simplified diagram to think about the heat diffusion demo. We have two 00:08:57.200 --> 00:09:04.140 bars made of the same material. One bar is hot and one is cold. We'll look at 4 atoms 00:09:04.140 --> 00:09:10.490 making up each bar. The hot bar has more energy than the cold bar -- its atoms are moving 00:09:10.490 --> 00:09:14.640 more than the atoms in the cold bar, which seem barely to move. 00:09:14.640 --> 00:09:19.890 Now, before we put the cold bar in contact with the hot bar, let's think about each bar 00:09:19.890 --> 00:09:25.100 separately. In our simplified drawing of the cold bar, let's say that three of the atoms 00:09:25.100 --> 00:09:30.520 have no energy and one atom has one quantum of energy and is at a slightly higher energy 00:09:30.520 --> 00:09:35.930 level, symbolized by the set of curved lines representing its motion. How many different 00:09:35.930 --> 00:09:40.660 microstates can this system exhibit? If we think about the different ways we can 00:09:40.660 --> 00:09:45.890 distribute the quantum of energy amongst the 4 atoms, we see that there are 4 possible 00:09:45.890 --> 00:09:51.010 microstates. If we do the same for our hot bar, where we 00:09:51.010 --> 00:09:56.560 have 5 quanta of energy that can be distributed in a variety of ways amongst the 4 atoms, 00:09:56.560 --> 00:10:02.360 we use some math to see that there are 56 possible microstates. 00:10:02.360 --> 00:10:06.570 When we brought the two bars in contact in our demonstration, we saw that they reached 00:10:06.570 --> 00:10:11.650 thermal equilibrium. Here, in our simplified example, we will bring 00:10:11.650 --> 00:10:17.780 the cold bar (defined as our system) and the hot bar (defined as our surroundings) together 00:10:17.780 --> 00:10:23.460 and divide the 6 quanta of energy equally between the two. The first law of thermodynamics 00:10:23.460 --> 00:10:27.520 tells us that the total of 6 quanta will be conserved. 00:10:27.520 --> 00:10:34.520 Now, how many microstates are now possible in each bar? 00:10:35.250 --> 00:10:39.880 As you might have expected, the number of possible microstates in what was originally 00:10:39.880 --> 00:10:44.930 our hot bar decreased, and the number of possible microstates in what was originally our cold 00:10:44.930 --> 00:10:48.400 bar increased. Let's see what this means for our total entropy 00:10:48.400 --> 00:10:55.400 change. We will use a relationship for entropy that was derived by Ludwig Boltzmann. It states 00:10:56.020 --> 00:11:00.810 that entropy is equal to a constant, called the Boltzmann constant, times the natural 00:11:00.810 --> 00:11:03.720 log of the number of microstates. 00:11:03.720 --> 00:11:09.040 When calculating entropy change, whether it be for the system or surroundings, delta S 00:11:09.040 --> 00:11:15.080 would be equal to Boltzmann's constant times the natural log of the ratio of the final 00:11:15.080 --> 00:11:19.630 number of microstates to the initial number of microstates. 00:11:19.630 --> 00:11:24.550 The entropy change in our cold bar was positive while the entropy change in our hot bar was 00:11:24.550 --> 00:11:30.970 negative. But remember, it's the total entropy change that matters. We see that our total 00:11:30.970 --> 00:11:36.080 entropy change for this process is positive. The spontaneous transfer of heat from our 00:11:36.080 --> 00:11:42.690 hot bar to our cold bar is consistent with the 2nd law of thermodynamics. 00:11:42.690 --> 00:11:47.240 If you did a similar calculation for the reverse process, that of heat transferring from the 00:11:47.240 --> 00:11:52.130 cold bar to the hot bar, the total entropy change would be negative indicating that it 00:11:52.130 --> 00:11:57.110 is not spontaneous. As we hinted earlier and as you may have guessed 00:11:57.110 --> 00:12:02.240 by our very simplified scenario, calculating the number of microstates in a real system 00:12:02.240 --> 00:12:07.940 can be very challenging. Generally speaking, you will be calculating entropy in terms of 00:12:07.940 --> 00:12:14.740 measurable macroscopic quantities such as heat capacity or enthalpy of phase change. 00:12:14.740 --> 00:12:19.940 However, having a qualitative understanding of the physical meaning of entropy will help 00:12:19.940 --> 00:12:24.930 you properly interpret the entropy changes caused by various processes. 00:12:24.930 --> 00:12:31.760 To Review, for a process to proceed spontaneously, the total entropy change for a system and 00:12:31.760 --> 00:12:37.070 its surroundings must be positive. Entropy measures the number of possible ways energy 00:12:37.070 --> 00:12:43.920 can be distributed in a system of molecules. A microstate is an instantaneous catalog that 00:12:43.920 --> 00:12:49.870 describes the energy of each molecule in a system. Because molecules are constantly interacting 00:12:49.870 --> 00:12:56.870 and exchanging energy, this description constantly needs to be revised. A given system has a 00:12:56.950 --> 00:13:03.060 large number of possible microstates. As we saw with the Boltzmann equation, entropy is 00:13:03.060 --> 00:13:10.060 proportional to the number of microstates.