WEBVTT 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 00:00:11.389 --> 00:00:14.789 of the circuit is made from copper foil. 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 00:00:20.029 --> 00:00:24.939 thermal expansion properties of copper, we can place some candles under the foil to provide 00:00:24.939 --> 00:00:31.939 enough heat to expand the foil and complete the circuit again! Many materials, like metals, 00:00:32.200 --> 00:00:34.100 expand when you heat them. 00:00:34.100 --> 00:00:37.730 But other materials, like polymers, shrink when heated. The macroscopic properties of 00:00:37.730 --> 00:00:39.530 both of these materials are highly dependent on temperature. 00:00:39.530 --> 00:00:44.730 The difference in the macroscopic behavior of these two materials is determined by very 00:00:44.730 --> 00:00:51.730 different microscopic structure. Statistical Mechanics is the method used to describe and 00:00:53.260 --> 00:01:00.260 predict behavior at the macro-scale based on statistical models of microscopic behavior. 00:01:00.780 --> 00:01:06.060 The first step to understanding the power of Statistical Mechanics, is to use this method 00:01:06.060 --> 00:01:13.060 to define and understand temperature in terms of macro state parameters. The notion of equilibrium 00:01:13.890 --> 00:01:18.110 will also prove to be useful in this exercise. 00:01:18.110 --> 00:01:21.230 This video is part of the equilibrium video series. 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, 00:01:27.580 --> 00:01:34.030 we must understand how a system's equilibrium state is constrained by its boundary and surroundings. 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 00:01:40.090 --> 00:01:41.700 and Engineering. 00:01:41.700 --> 00:01:47.400 Before watching this video, you should be familiar with the second law of thermodynamics, 00:01:47.400 --> 00:01:53.670 and the definitions of micro-state, macro-state, and entropy. 00:01:53.670 --> 00:01:59.030 After watching this video, you will be able to: Explain how the definition of temperature 00:01:59.030 --> 00:02:06.030 arises as a derivative of entropy with respect to energy. 00:02:09.250 --> 00:02:16.250 Consider an isolated, insulated box of non-interacting particles. We can characterize this box macroscopically 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 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 00:02:28.810 --> 00:02:35.810 total energy in this box are fixed. Thus a microstate that corresponds to such a macrostate 00:02:35.849 --> 00:02:41.670 is the velocity and position {vi, xi} of each particle in the box. With this information, 00:02:41.670 --> 00:02:46.620 we can determine the movement and position of every particle in the box, and we obtain 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 00:02:55.349 --> 00:03:01.840 in the system by tracking each and every microstate would be a computational nightmare. 00:03:01.840 --> 00:03:08.560 Instead, we consider Omega, the number of microstates that correspond to a system with 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 00:03:17.040 --> 00:03:24.040 of non-interacting particles. The wall separating the two boxes allows the transfer energy, 00:03:24.150 --> 00:03:29.599 but does not allow particles to cross it. What changes in macro state parameters do 00:03:29.599 --> 00:03:36.599 we observe in our system? Pause the video here. The volume and number of particles in 00:03:41.879 --> 00:03:48.519 each box is fixed and unchanging. Because energy, in the form of heat, can be transferred 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 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 00:04:03.709 --> 00:04:09.599 microstates that the composite system can have, which correspond to a total system energy 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 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 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 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 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 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. 00:04:54.320 --> 00:04:59.030 Let's look at a typical example to see what f looks like. 00:04:59.030 --> 00:05:04.150 We can explore the function f by considering a simple case, where each particle can only 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 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 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 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 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 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 00:05:50.150 --> 00:05:56.080 that the sum that defines omega is the area under this graph. 00:05:56.080 --> 00:06:03.080 Look at how peaked this graph is! With increasing numbers of particles, the graph becomes more 00:06:03.310 --> 00:06:10.310 and more strongly peaked. Remember, in a realistic situation, the number of particles is going 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 00:06:16.150 --> 00:06:23.150 highly peaked with a very small standard deviation. Statistically, this means as the Number of 00:06:23.650 --> 00:06:29.770 particles becomes larger, the energy in box 1 is almost always very close to this peak 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 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 00:06:53.340 --> 00:06:55.630 f with respect to E1. 00:06:55.630 --> 00:07:01.080 Since the volume, number, and total energy of the system are fixed, differentiate f with 00:07:01.080 --> 00:07:06.770 respect to E1. The maximum occurs when this derivative is zero. 00:07:06.770 --> 00:07:11.720 Take a moment to carry out this derivative, and check your solution with ours. Pause the 00:07:11.720 --> 00:07:18.720 video. 00:07:19.440 --> 00:07:26.440 We obtain the following expression. Thus the Energy Em that leads to the greatest number 00:07:26.960 --> 00:07:32.610 of composite system microstates is defined by this elegant condition. 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. 00:07:40.669 --> 00:07:46.800 We'll explain why we include it later. By defining this new term, entropy, our condition 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 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 00:07:59.039 --> 00:08:05.960 2. 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 00:08:13.520 --> 00:08:18.819 if box 1 starts with some different energy, maybe an energy significantly different from 00:08:18.819 --> 00:08:25.819 Em? Statistically, because such a state is so much less likely, when contact between 00:08:25.880 --> 00:08:32.209 the boxes is made, the energy will redistribute over time towards the most likely state, where 00:08:32.209 --> 00:08:35.349 box 1 has energy Em. 00:08:35.349 --> 00:08:42.139 This process of energy redistribution is called equilibration. And the state with the highest 00:08:42.139 --> 00:08:45.449 likelihood is called equilibrium! 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 00:08:50.899 --> 00:08:57.329 earlier, evaluated at the average energy of each box is held, it indicates that we are 00:08:57.329 --> 00:09:04.329 at equilibrium. Note that in the simple example scenario we considered earlier, it was NOT 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 00:09:09.999 --> 00:09:15.879 respect to energy for each box evaluated at the energy, number, and volume of each box 00:09:15.879 --> 00:09:20.189 that must be equal at equilibrium. 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 00:09:26.470 --> 00:09:33.470 new system variable. But what would this variable represent physically? 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 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 00:09:46.779 --> 00:09:53.779 allows energy to transfer. Over time, what parameter will eventually be the same for 00:09:53.970 --> 00:10:00.970 both boxes? Pause the video and discuss. 00:10:04.160 --> 00:10:09.410 Our experience tells us that once 2 subsystems are brought into thermal contact, we expect 00:10:09.410 --> 00:10:15.959 the composite system will eventually evolve so that each box has the same temperature! 00:10:15.959 --> 00:10:21.249 But at equilibrium the derivative of entropy with respect to energy of each box is also 00:10:21.249 --> 00:10:28.249 equal. This tells us that this derivative should be some function of temperature. In 00:10:28.389 --> 00:10:34.110 fact, the derivative of entropy with respect to energy is exactly equal to the reciprocal 00:10:34.110 --> 00:10:40.519 of temperature! This tells us that when we are measuring temperature of a system, we 00:10:40.519 --> 00:10:45.989 are NOT measuring the energy of a system, we are measuring this derivative! 00:10:45.989 --> 00:10:51.309 There are specific units determined by the typical way we measure temperature. This is 00:10:51.309 --> 00:10:56.910 why Boltzmann's constant was introduced into the definition of entropy! The constant is 00:10:56.910 --> 00:11:02.980 introduced precisely so that the derivative of entropy with respect to energy has dimension 00:11:02.980 --> 00:11:09.980 of 1/temperature. 00:11:12.279 --> 00:11:18.410 In this video we've seen that entropy is a natural macrostate parameter, and statistically 00:11:18.410 --> 00:11:24.639 a system's microstates evolve to exist within the maximum entropy state. This is called 00:11:24.639 --> 00:11:31.519 equilibrium. Temperature is naturally defined as the derivative of entropy with respect 00:11:31.519 --> 00:11:37.619 to energy for a system. And this definition allows us to understand how temperature and 00:11:37.619 --> 00:11:44.619 entropy are related at equilibrium.