PHYS 172: Modern Mechanics Lecture 23 – Heat Capacity Read 12.6 Summer 2012

Last Time Einstein Model of Solids (springs + balls) # microstates N oscillators & q quanta Fundamental assumption of statistical mechanics Over time, an isolated system in a given macrostate (total energy) is equally likely to be found in any of its microstates (microscopic distribution of energy).

Energy is exchanged until the most probable distribution is reached. Equilibrium = Most Probable Distribution

Clicker Question There is thermal transfer of energy of 5000 J into a system. The entropy of the system increases by 50 J/K. What is the approximate temperature of the system? 1) 5000 K 2) 100 K 3) 50 K 4) 0.01 K 5) K ANSWER:

Brief Review of Heat Capacity Today: Heat Capacity Pb vs Al: A Chain of Reasoning Quantum versus Classical

Note: How Do Heat and Work Differ? WORK Compress a solid (force through a distance): SHAPE changes  Energy Levels Change HEAT Energy levels don’t change. Transfer quanta from one place to another.

Heat Capacity How much heat do you have to add to change the temperature by a certain amount? a)Large amount  Large heat capacity b)Small amount  Small heat capacity Has to do with degrees of freedom -- where are all the microscopic places the system can store energy. More modes = higher heat capacity. (Energy will go into every mode it can...)

Heat Capacity How much heat do you have to add to change the temperature by a certain amount? (Heat = energy transferred) Water has a high heat capacity. Live near water.

Heat Capacity of Solids Which has the higher heat capacity, Lead (Pb) or Aluminum (Al)? Lead Aluminum

Each block contains 6 x atoms. m Al = 27gm Pb = 207g Same crystal structure. Which shows the right relative sizes? A) B) C) Which is Bigger? ANSWER: Diameters of all atoms are roughly the same. The difference in mass is due to the mass of the nuclei (which in either case is just a “dot” at the center of the atom)

Initially the two blocks are at a temperature very near absolute zero (0 K). We will add 1 J of energy to the aluminum block, and 1 J of energy to the lead block, and see which block has the larger increase in temperature. We will step through a chain of reasoning using statistical mechanics to answer this question, which will let us determine whether aluminum or lead has the higher heat capacity at low temperatures. Heat Capacity for Pb and Al

CLICKER QUESTION Einstein model = independent quantum harmonic oscillators. Which shows the right energy level diagram? A) Al Pb B) Pb Al ANSWER: k Al = 16 N/mk Pb = 5 N/m m Al = 27gm Pb = 207g(1 mole of each)

CLICKER QUESTION We add 1 J of energy to each block. Given the fact that Al has the greater energy-level spacing, which block now has the larger number of quanta of energy, q? A) Al B) Pb C) Same for both. 4.0e-21 J Answer: Al has 1 quantaPb has ≈ 5 quanta Note: Energy spacing of levels is not shown correctly for this well!

CLICKER QUESTION Which block has the largest number N of quantized oscillators? A) N is greater in the Al B) N is greater in the Pb C) N is the same for Pb and Al Both blocks have 1 mole = 6 x atoms = 18 x oscillators. ANSWER:

CLICKER QUESTION The Pb block has more quanta corresponding to the 1J of thermal energy. Therefore, in which block is there a larger number of ways Ω of arranging the thermal energy? A) The number of ways Ω is greater in the Al B) The number of ways Ω is greater in the Pb C) The number of ways Ω is the same in the Pb and the Al # microstates N oscillators & q quanta

CLICKER QUESTION The Pb block has the larger number of ways Ω to arrange the energy. So which block now has the larger entropy S? A) The entropy S is now greater in the Al B) The entropy S is now greater in the Pb C) The entropy S is the same in the Al and the Pb Entropy S = klnΩ is larger for Pb. ANSWER:

CLICKER QUESTION Originally the temperature of the blocks was near absolute zero, with almost no thermal energy in the blocks. How many ways are there to arrange zero energy in a block? Just 1. So what was the original entropy in a block? A) 0 J/K B) 1 J/K C) infinite (Remember 1 = x 0  ln1=0). Entropy S = klnΩ = kln1 = 0. ANSWER:

CLICKER QUESTION We found that after adding 1 J to each block, the entropy S is now greater in the Pb block. Both blocks started with zero entropy. Therefore which block experienced a larger change in entropy ΔS? A) The entropy change ΔS was greater in the Al B) The entropy change ΔS was greater in the Pb C) The entropy change ΔS was the same in the Pb and the Al

CLICKER QUESTION We added the same amount of energy ΔE = 1 J to each block, and the entropy change ΔS was greater in the Pb block. Which block now has the higher temperature? A) The temperature of the Al is now higher B) The temperature of the Pb is now higher C) The temperature of the Al and Pb are the same ANSWER: Thus, 1/T is bigger for Pb

CLICKER QUESTION The original temperature was 0 K, and the final temperature of the Al block is higher than that of the Pb block, so the Al block has the larger change in temperature, ΔT. At low temperatures, which block has the greater heat capacity per atom, C = (ΔE / ΔT) / 6e23? A) The low-temperature heat capacity per atom of Al is greater B) The low-temperature heat capacity per atom of Pb is greater C) Same for both. Measured heat capacities

Specific Heat and Quantization At high temperatures, average energy per oscillator is very high. Energy levels are packed so tight they look continuous. Classical (non-quantum) theory is a good approximation here! At low temperatures, energy quantization is really apparent: energies are nowhere near continuous. Classical theory fails badly here.

Specific Heat and Quantization classical expectation NOTE: The classical expectation was known to disagree with the data long ago. Quantization of energy levels solved this paradox, one of the first signs of the need for quantum mechanics.

Spring Model Limitations At high temperatures, average energy per oscillator is high. The non-uniform spacing of the levels means that the spring-model is breaking down. Thus, the Einstein model starts to break down at high temperatures.

Improving Einstein’s Model Are atoms in a solid really isolated from each other? No! Really: atoms interact with each other. Atoms don’t oscillate independently. They have waves called phonons. How to be better than Einstein: Treat the phonons as harmonic oscillators! You’ll see this in Physics 416.

Brief Review of Specific Heat Today: Specific Heat Capacity Pb vs Al: A Chain of Reasoning Quantum versus Classical Derivation Next Lecture: Boltzmann Distribution Application: Kinetic Theory of Gasses