Reading Quiz

Question 1:

Finally, finally, finally, we have a robust definition of temperature. What is temperature, according to this reading?

Answer:

We can see, according to eq. (3.5) that the reciprocal of the temperature is given by the partial derivative of the entropy of the system with respect to its energy, while holding N, V fixed.
  1. The reciprocal of the slope of a system's entropy vs. evergy graph (while the volume and number of molecules are held fixed)
  2. "The temperature of a system is the reciprocal of the slope of its entropy vs. engergy graph," (pg 88) which can be obtained by taking the partial derivative of an equation of S with respect to U of some equation S(U) while holding N and V constant.
  3. The temperature of a system is the reciprocal of the slope of its entropy vs. energy graph.
  4. It is the reciprocal of the slope of the temperature vs. energy graph at constant volume and constant number of particles.
  5. The reciprocal of the slope of its entropy vs. energy graph.
  6. Temperature of a system is the reciprocal of the partial derivative of entropy with respect to energy while holding volume and number of particles fixed.
  7. Temperature is the derivative of the energy with respect to entropy, with number of particles and volume held constant. T = (dU/dS)_{N,V}
  8. Temperature is the change in energy divided by the change in entropy
  9. Specifically, the reciprocal of the slope of the entropy vs. energy graph for a system (if N, V are held constant). That is, it describes the way entropy changes with increasing or decreasing energy.
  10. Temperature is the inverse of the slope of the entropy vs. energy graph for a system.
  11. When two objects in thermal contact reach the highest entropy value possible for their system.

Question 2:

In Schroeder's "silly analogy", what happens to the temperature of the "normal" people when you add energy? What happens to the temperature of the "miserly" people when you add energy? What strange thing do you notice about the temperature of the "enlightened" people? (It may help to look at Figure 3.2 to help answer this question).

Answer:

Looking at the slopes in Figure 3.2, we see that as you add energy to the "normal" people, the slope of S vs. U becomes smaller, which means the temperature increases. When you add energy to the "miserly" people, the slope of S vs. U becomes larger, which means the temperature decreases. If you look at the slope of S vs. U for the "enlightened" people, you notice that it is always NEGATIVE, which means that this system has a NEGATIVE TEMPERATURE?!?! We'll discuss this more in section 3.3.
  1. When you add energy to the normal people, their temperature (or generosity) goes up. Miserly people's temperature decreases as you add energy. The enlightened people are such that they don't want to gain energy, and the entropy of the system decreases as you increase energy.
  2. Normal people: when energy is added, temperature increases. miserly: when energy is added, temperature decreases The energy of "enlightened" people decreases as entropy increases
  3. As you add energy, the temperature of "normal" people increases. With "miserly" people, the temperature decreases. "Enlightened" people have an increase in temperature as they lose energy.
  4. Normal people become hotter when you add energy. Miserly people become colder when you add energy. The slope decreases as energy decreases, so the temperature increases as you remove energy? Odd.
  5. The temperature of the normal people increases when you add energy. The temperature of the miserly people decreases as energy is added. The temperature of the enlightened people increases as they lose energy.
  6. As you add energy: "normal" people - temperature increases "miserly" people - temperature decreases As you remove energy: "enlightened" people - temperature increases
  7. When you add energy to "normal" people, they get happier/hotter. When you add energy to "miserly" people, they get less happy/colder. The "enlightened" people get happier/hotter when they LOSE energy.
  8. The normal people's temperature increases when they get energy, but the miserly people undergo a temperature drop when given energy. The enlightened system seems to not even want energy...
  9. The normal people increase the temperature when you increase energy. For miserly people, this is the opposite, they drop in temp. as they gain energy. For enlightened people it seems that the temparature would stay the same (?)
  10. Normal people get hotter as they gain energy, and miserly people get colder as the gain energy. As enlightened people gain energy, they become hotter, but their temperature is negative, so its going from a large negative value to a smaller negative value.
  11. the get a little happier they get a lot happier enlighted people will have a higher temperature since their generosity is higher

Question 3:

Please look at Figure 3.3 in your book. Assume that the scales on both graphs are the same, and that UA,initial = UB,initial = U. Which system (A or B) is at the higher temperature at the indicated U? Briefly explain your reasoning.

Answer:

Just look at the slopes. On an entropy vs. energy graph, the slope is the reciprocal of the temperature. Since the slope of the B graph is smaller than the slope of the A graph at the same point, the B is at the higher temperature.
  1. The graph for A has a bigger slope than the graph for B. Thus since temperature is the inverse of the entropy versus energy graph graph B has a higher energy.
  2. temperature=reciprocal of slope Slope A steeper then slope B Slope A and slope B both < 1 Slope B is at higher temperature.
  3. System B is at the higher temperature at whichever U you are looking at because the slope of its entropy vs. energy graph is shallower than A's.
  4. The slope for A is larger than for B, so system A must be at the lower temperature (due to the inverse relationship between temperature and slope of the graph).
  5. System B is at the higher temperature. In both situations U is the same but S is greater in A. Since S is greater and S has an inverse relationship to Temperature, the temperature of A is less than that of B.
  6. System B has a higher temperature because the dS/dU is smaller for B, thus the reciprocal of this is bigger, and this is the definition of temperature.
  7. System B is at a higher temperature, since the slope of the entropy vs. energy graph is less steep than that of system A.
  8. System A has a higher temperature. According to 3.6 we can infer that the temperature is the slope of the energy/ entropy graph at a specific U. In this case the A system has a higher slope.
  9. Well A has a higher instantaneous dS/DU. however, since T = (ds/du)^-1, it means that A has a smaller T, and so B is at the higher temperature at the indicated U.
  10. System B is at a higher temperature, since the slope of its graph is smaller (closer to the horizontal) than System B.
  11. Sb, because the system is at a lower probability, high energy state, which means that the temperature is higher.

Question 4:

What material from the reading or previous classes would you like me to go over in more detail?

Answer:

Your responses below.
  3. As for Figure 3.1, does the point where S_A and S_B cross mean anything?
  5. This section seemed pretty easy to handle. A quick overview of the silly analogy could be helpful including what kind of objects would be represented by the miserly and enlightened people.
  7. Anything that's (supposed to be) on page 86 in the book. My page 86 is part of a set of problems from Chapter 7.
  8. nohin
  9. The analogy portion was a little confusing.
  10. What in the world would an "Enlightened" System be like? Would it have to have negative temperature?
  11. ahh this analogy makes me feel weird.