Internal energy's natural variables are entropy, S, and V. Mathematically, it can be written as U=U(S,V). Write the total differential of U. Derive the Maxwell relation from the total differential and evaluate the partial differentials.
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ACTIVITY 2-4 part 1
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Solved in 4 steps
- 05. Find the total number of microstates of distributing 5 distinguishable particles among energy level 0, E, 2E, 3E, and 4E such that the internal Energy remains 4E. a. How many distributions are possible? b. Find the total number of microstates c. What is the most probable distribution? d. What is the Entropy of the system? Please make a table clearly showing the energy levels and particle distribution. Show your calculations.j. Define the relaxation time. Write all mathematical formulas needed to explain the definition.Problem 1: Thermodynamics of neutral Fermi liquid: a) Write the free energy of quasi-particles in the theory of Fermi liquid. Discuss the difference of quasi-particles free energy and the free energy of non-interacting Fermi gas. b) What are the characteristic energy scales in the theory of Fermi liquid? c) What is the difference in the specific heat of quasi-particles in the theory of Fermi liquids and the specific heat of non-interacting Fermi gas. Plot the specific heat vs. temperature for both systems.
- 1. Define packing fraction.mention some example? 2. The primitive cells in reciprocal space is known as the.........?1. N independent simple harmonic oscillators.2. Differential of a function of several variables.3. First law of thermodynamics - recap.4. Chemical potential defined. Re-write the first law of thermodynamics expressing S as a function of U, V, N.Now explain how the pressure, temperature and chemical potential may be computedif somehow we were given the entropy.For each of these problems, think about and then describe how you break the problem down to be able to solve it. Then do the Fermi estimate. 1. Estimate the number of pieces of paper that would be needed to cover the surface of the earth. 2. How long a line can you draw with a pencil?
- Problem 1: Rubber bands and Universality Polymers, like rubber, are made of very long molecules, usually tangled up in a configuration with lots of entropy. As a crude model of a rubber band, consider a chain of N links, each of length e (see below). Imaging that each link has only two possible states, pointing either left or right. The total length L of the rubber band is the net displacement from the beginning of the first link to the end of the last link. N links a) Find an expression for the entropy of this system in terms of N and Ng, the number of links pointing to the right. b) Write down a formula for L in terms of N and Ng. c) For a one-dimensional system such as this, the length L is analogous to the volume V of a three-dimensional system. Similarly, the pressure P is replaced by the tension force F. Taking F to be positive when the rubber band is pulling inward, write down and explain the appropriate thermodynamic identity for this system. d) Using the thermodynamic…sieice Please derive the 2D density of states and plot the curve of density of states(DOS) versus energy, and explain physical property of DOS. D(E) EConsider Ω = αU^(αNV) Use the following steps to find T, U(T), and CV 1. Use quantum mechanics and some combinatorics to find an expression for the multiplicity,Ω , in terms of U, V, N, and any other relevant variables. 2. Take the logarithm to find the entropy, S. 3. Differentiate S with respect to U and take the reciprocal to find the temperature, T, as a function of U and other variables. 4. Solve for U as a function of T (and other variables). 5. Differentiate U(T) to obtain a prediction for the heat capacity (with the other variables held fixed).
- Kinetic energy of electron gas. Show that the kinetic energy of a three-dimensional gas of N free electrons at 0 K is U, = N€p .Why is the probability of a system's ground state larger than the probabilities of its excited states for a system in thermal equilibrium with a reservoir? In answering this you may want to consider your calculations in this worksheet. a. Adding energy to the large reservoir increases the total entropy more than adding the same amount of energy to the system. b. Removing all energy from the system increases its entropy the most. c. The probability of the state of the system is maximized when the number of microstates of the system is maximized. d. The probability of the state of the system is maximized when the number of microstates for that state of the system and reservoir is maximized.1. Compute the partition function Z¡(T, V, N, B) of a single molecule as a function of the temperature T, volume V, the number of molecules N, and the magnetic field B. 2. What are the probabilities for S of a specific molecule to take the values +1 and – 1 at temperature T? 3. Find the pressure P of the gas as a function of T, V, N, B. How the pressure behaves as you flip the direction of the magnetic field?