1 00:00:06,220 --> 00:00:11,389 Here you see a circuit powered by a battery and connected through a light bulb. This section 2 00:00:11,389 --> 00:00:14,789 of the circuit is made from copper foil. 3 00:00:14,789 --> 00:00:20,029 We can put a cut into the foil to make this circuit incomplete. Taking advantage of the 4 00:00:20,029 --> 00:00:24,939 thermal expansion properties of copper, we can place some candles under the foil to provide 5 00:00:24,939 --> 00:00:31,939 enough heat to expand the foil and complete the circuit again! Many materials, like metals, 6 00:00:32,200 --> 00:00:34,100 expand when you heat them. 7 00:00:34,100 --> 00:00:37,730 But other materials, like polymers, shrink when heated. The macroscopic properties of 8 00:00:37,730 --> 00:00:39,530 both of these materials are highly dependent on temperature. 9 00:00:39,530 --> 00:00:44,730 The difference in the macroscopic behavior of these two materials is determined by very 10 00:00:44,730 --> 00:00:51,730 different microscopic structure. Statistical Mechanics is the method used to describe and 11 00:00:53,260 --> 00:01:00,260 predict behavior at the macro-scale based on statistical models of microscopic behavior. 12 00:01:00,780 --> 00:01:06,060 The first step to understanding the power of Statistical Mechanics, is to use this method 13 00:01:06,060 --> 00:01:13,060 to define and understand temperature in terms of macro state parameters. The notion of equilibrium 14 00:01:13,890 --> 00:01:18,110 will also prove to be useful in this exercise. 15 00:01:18,110 --> 00:01:21,230 This video is part of the equilibrium video series. 16 00:01:21,230 --> 00:01:27,580 It is often important to determine whether or not a system is at equilibrium. To do this, 17 00:01:27,580 --> 00:01:34,030 we must understand how a system's equilibrium state is constrained by its boundary and surroundings. 18 00:01:34,030 --> 00:01:40,090 Hi, my name is Jeff Grossman, and I'm a professor in the MIT Department of Materials Science 19 00:01:40,090 --> 00:01:41,700 and Engineering. 20 00:01:41,700 --> 00:01:47,400 Before watching this video, you should be familiar with the second law of thermodynamics, 21 00:01:47,400 --> 00:01:53,670 and the definitions of micro-state, macro-state, and entropy. 22 00:01:53,670 --> 00:01:59,030 After watching this video, you will be able to: Explain how the definition of temperature 23 00:01:59,030 --> 00:02:06,030 arises as a derivative of entropy with respect to energy. 24 00:02:09,250 --> 00:02:16,250 Consider an isolated, insulated box of non-interacting particles. We can characterize this box macroscopically 25 00:02:17,230 --> 00:02:23,099 by 3 parameters: the Volume of the box, the Number of particles in the box (measured in 26 00:02:23,099 --> 00:02:28,810 moles), and the total Energy. The Volume of the box, and the Number of particles and the 27 00:02:28,810 --> 00:02:35,810 total energy in this box are fixed. Thus a microstate that corresponds to such a macrostate 28 00:02:35,849 --> 00:02:41,670 is the velocity and position {vi, xi} of each particle in the box. With this information, 29 00:02:41,670 --> 00:02:46,620 we can determine the movement and position of every particle in the box, and we obtain 30 00:02:46,620 --> 00:02:53,620 the total energy is ½ mv^2 summed over all particles. As you can imagine, measuring changes 31 00:02:55,349 --> 00:03:01,840 in the system by tracking each and every microstate would be a computational nightmare. 32 00:03:01,840 --> 00:03:08,560 Instead, we consider Omega, the number of microstates that correspond to a system with 33 00:03:08,560 --> 00:03:15,560 macro-parameters E, N, and V. Now, let's consider an isolated system comprised of two boxes 34 00:03:17,040 --> 00:03:24,040 of non-interacting particles. The wall separating the two boxes allows the transfer energy, 35 00:03:24,150 --> 00:03:29,599 but does not allow particles to cross it. What changes in macro state parameters do 36 00:03:29,599 --> 00:03:36,599 we observe in our system? Pause the video here. The volume and number of particles in 37 00:03:41,879 --> 00:03:48,519 each box is fixed and unchanging. Because energy, in the form of heat, can be transferred 38 00:03:48,519 --> 00:03:55,519 through the wall, E1 and E2 can change. But of course, the total energy E = E1 + E2 stays 39 00:03:57,440 --> 00:04:03,709 the same, because the 2-box system is thermally isolated. Let's count the total number of 40 00:04:03,709 --> 00:04:09,599 microstates that the composite system can have, which correspond to a total system energy 41 00:04:09,599 --> 00:04:16,599 E. The easiest way to do this is to first define a new function f(E1), which is the 42 00:04:16,889 --> 00:04:23,889 number of microstates of the composite system when box 1 has energy E1. To do this, we count 43 00:04:24,290 --> 00:04:30,430 the number of microstates where box 1 has some energy E1, and multiply this by the number 44 00:04:30,430 --> 00:04:37,430 of microstates where box 2 has energy E-E1. Then, to find the total number of microstates 45 00:04:40,000 --> 00:04:46,120 for the composite system, we need to sum the function f over all ways we could have selected 46 00:04:46,120 --> 00:04:53,120 E1. We want to understand more about this function f because it is key to defining Omega. 47 00:04:54,320 --> 00:04:59,030 Let's look at a typical example to see what f looks like. 48 00:04:59,030 --> 00:05:04,150 We can explore the function f by considering a simple case, where each particle can only 49 00:05:04,150 --> 00:05:11,150 exist in one of 2 allowable energy states—one with energy 0 and the other with 1. Suppose 50 00:05:12,910 --> 00:05:19,560 I have 50 particles in box 1 and 100 particles in box 2, and the total energy of the system 51 00:05:19,560 --> 00:05:26,560 is fixed at, say, 50. In order to reach this total energy, the energy E1 in box 1 can range 52 00:05:27,400 --> 00:05:34,400 from 0 to 50. This is the graph for the number of microstates with total energy E having 53 00:05:34,460 --> 00:05:41,460 a given E1. We get this graph by looking at all ways to assign every particle a 0 or 1 54 00:05:42,130 --> 00:05:49,130 state such that the energy in box 1 is E1 and the energy in box 2 is E-E1. We can imagine 55 00:05:50,150 --> 00:05:56,080 that the sum that defines omega is the area under this graph. 56 00:05:56,080 --> 00:06:03,080 Look at how peaked this graph is! With increasing numbers of particles, the graph becomes more 57 00:06:03,310 --> 00:06:10,310 and more strongly peaked. Remember, in a realistic situation, the number of particles is going 58 00:06:10,509 --> 00:06:16,150 to be on the order of Avagadro's number. Which means the distribution of energies will be 59 00:06:16,150 --> 00:06:23,150 highly peaked with a very small standard deviation. Statistically, this means as the Number of 60 00:06:23,650 --> 00:06:29,770 particles becomes larger, the energy in box 1 is almost always very close to this peak 61 00:06:29,770 --> 00:06:36,770 value. Let's define Em to be the box 1 energy corresponding to the peak value of f. So how 62 00:06:40,270 --> 00:06:47,270 do we find Em? Pause the video and think about it. To find Em, we want to maximize the function 63 00:06:53,340 --> 00:06:55,630 f with respect to E1. 64 00:06:55,630 --> 00:07:01,080 Since the volume, number, and total energy of the system are fixed, differentiate f with 65 00:07:01,080 --> 00:07:06,770 respect to E1. The maximum occurs when this derivative is zero. 66 00:07:06,770 --> 00:07:11,720 Take a moment to carry out this derivative, and check your solution with ours. Pause the 67 00:07:11,720 --> 00:07:18,720 video. 68 00:07:19,440 --> 00:07:26,440 We obtain the following expression. Thus the Energy Em that leads to the greatest number 69 00:07:26,960 --> 00:07:32,610 of composite system microstates is defined by this elegant condition. 70 00:07:32,610 --> 00:07:39,610 So let's define a new macro-parameter, S, called "entropy". The kB is Boltzmann's constant. 71 00:07:40,669 --> 00:07:46,800 We'll explain why we include it later. By defining this new term, entropy, our condition 72 00:07:46,800 --> 00:07:53,020 above occurs when the derivative of the entropy of box 1 with respect to the energy of box 73 00:07:53,020 --> 00:07:59,039 1 is equal to the derivative of the entropy of box 2 with respect to the energy of box 74 00:07:59,039 --> 00:08:05,960 2. 75 00:08:05,960 --> 00:08:12,960 So on average, we expect the energy of box 1 to be very close to Em. But what happens 76 00:08:13,520 --> 00:08:18,819 if box 1 starts with some different energy, maybe an energy significantly different from 77 00:08:18,819 --> 00:08:25,819 Em? Statistically, because such a state is so much less likely, when contact between 78 00:08:25,880 --> 00:08:32,209 the boxes is made, the energy will redistribute over time towards the most likely state, where 79 00:08:32,209 --> 00:08:35,349 box 1 has energy Em. 80 00:08:35,349 --> 00:08:42,139 This process of energy redistribution is called equilibration. And the state with the highest 81 00:08:42,139 --> 00:08:45,449 likelihood is called equilibrium! 82 00:08:45,449 --> 00:08:50,899 How do we know we've landed in an equilibrium state? If the derivative condition we found 83 00:08:50,899 --> 00:08:57,329 earlier, evaluated at the average energy of each box is held, it indicates that we are 84 00:08:57,329 --> 00:09:04,329 at equilibrium. Note that in the simple example scenario we considered earlier, it was NOT 85 00:09:04,660 --> 00:09:09,999 the case that the energy of each box was the same. It is the derivative of entropy with 86 00:09:09,999 --> 00:09:15,879 respect to energy for each box evaluated at the energy, number, and volume of each box 87 00:09:15,879 --> 00:09:20,189 that must be equal at equilibrium. 88 00:09:20,189 --> 00:09:26,470 This suggests that we should define the derivative of entropy with respect to energy to be a 89 00:09:26,470 --> 00:09:33,470 new system variable. But what would this variable represent physically? 90 00:09:33,670 --> 00:09:40,300 To figure out what it should be, think about our composite system. The volume of each box 91 00:09:40,300 --> 00:09:46,779 is constant as is the number of particles in each box. The wall between the two boxes 92 00:09:46,779 --> 00:09:53,779 allows energy to transfer. Over time, what parameter will eventually be the same for 93 00:09:53,970 --> 00:10:00,970 both boxes? Pause the video and discuss. 94 00:10:04,160 --> 00:10:09,410 Our experience tells us that once 2 subsystems are brought into thermal contact, we expect 95 00:10:09,410 --> 00:10:15,959 the composite system will eventually evolve so that each box has the same temperature! 96 00:10:15,959 --> 00:10:21,249 But at equilibrium the derivative of entropy with respect to energy of each box is also 97 00:10:21,249 --> 00:10:28,249 equal. This tells us that this derivative should be some function of temperature. In 98 00:10:28,389 --> 00:10:34,110 fact, the derivative of entropy with respect to energy is exactly equal to the reciprocal 99 00:10:34,110 --> 00:10:40,519 of temperature! This tells us that when we are measuring temperature of a system, we 100 00:10:40,519 --> 00:10:45,989 are NOT measuring the energy of a system, we are measuring this derivative! 101 00:10:45,989 --> 00:10:51,309 There are specific units determined by the typical way we measure temperature. This is 102 00:10:51,309 --> 00:10:56,910 why Boltzmann's constant was introduced into the definition of entropy! The constant is 103 00:10:56,910 --> 00:11:02,980 introduced precisely so that the derivative of entropy with respect to energy has dimension 104 00:11:02,980 --> 00:11:09,980 of 1/temperature. 105 00:11:12,279 --> 00:11:18,410 In this video we've seen that entropy is a natural macrostate parameter, and statistically 106 00:11:18,410 --> 00:11:24,639 a system's microstates evolve to exist within the maximum entropy state. This is called 107 00:11:24,639 --> 00:11:31,519 equilibrium. Temperature is naturally defined as the derivative of entropy with respect 108 00:11:31,519 --> 00:11:37,619 to energy for a system. And this definition allows us to understand how temperature and 109 00:11:37,619 --> 00:11:44,619 entropy are related at equilibrium.