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- Given the internal energy U and entropy S of N weakly interacting particles in a closed system with fixed volume V. U = NkgT² (27In 2) U S = Nkg lnz + T (a) Prove the Helmholtz free energy (b) Prove the Pressure of the system is F = -NkBT ln z P = Nk Tln z) ( T(2nx sin \1.50. 2nz Consider the case of a 3-dimensional particle-in-a-box. Given: 4 = sin(ny) sin 2.00. What is the energy of the system? O 6h?/8m O 4h²/8m O 3h2/8m O none are correctRegion 1 is x 0 with a = 4&. If E2 = 6a - 10a, + 8a. V/m, (a) find Pi, and P2, (b) calculate the energy densities in both regions.
- The population ratio between two energy levels ni nj separated in energy by: A E = E₁ - Ej with AE = 1.1×10-22 J is 0.84. That is: ni = 0.84 with AE = 1.1×10-22] nj Remember the Boltzmann equation for the population of particles in state i with energy Ei at temperature T is: N n₁ = = e Z What is the temperature of the system (use two sig figs)? 4.0 ✓ KWhat is the partition function for the system shown? a. b. C. d. E₁ Eo -Eo/KT + e-E₁/kT 2e-Eo/kT +3e-ElkT 3e¯ e-Eo/2kT + e-E₁/3kT e-2Eo/kT + e-³E₁/KTIf the partition function is Z= VT and V=3 m^3, T=280 K, then the Enthalpy * :will be 1238.91 J 216931.566 J O 345.23 J O 415.77 J O
- 3) Consider the collection of identical harmonic oscillators (as in the Einstein floor). The permitted energies of each oscilator (E = nhf (n=0, 1, 2.0, hf. 2hf and so on. a) Calculate the splitting function of a single harmonic oscitor. What is the splitting function of N oscilator? wwww wwwwww www www b) Obtain the average energy of the T-temperature N oscilator from the split function. c) Calculate the heat capacity of this system and T → 0 ve T → 0 in limits, what is the heat capacity of the system? Are these results in line with the experiment? Why? What's the right theory about that? w w d) Find the Helmholtz free energy of this system. www ww e) which gives the entropy of this system as a function of temperature. ww wd wwww wwThe partition function of an ensemble at a temperature T is N Z = (2 cosh kgT where kg is the Boltzmann constant. The heat capacity of this ensemble at T = is X Nkg, where the value of X is %3D kB (up to two decimal places).When two galaxies collide, the stars do not generally run into eachother, but the gas clouds do collide, triggeering a burst of new star formation. a) Estimate the probability that our Sun would collide with another star in the Andromeda galaxy if a collision between the Milky Way and Andromeda occured. Assume that each galaxy has 100 billion stars exactly like the sun, spread evenly over a circular disk with a radius of 100,000 lightyears. (Hint: first calculate total area of 100 billion circles with the radius of the Sun and then compare that to total area to the area of the Galatic disk) b) Estimate the probability of a collision between a gas cloud in our galaxy and one in the Andromeda galaxy. Assume that each galaxy has 100,000 clouds of warm hydrogen gas, each with a radius of 300 lightyears, spread evenly over the same disk. Use the same method as part A.
- For a system of 4 distinguishable particles to be distributed in 6 allowed energy states namely E0 (0 eV), E1 (1 eV), E2 (2 eV), E3 (3 eV), E4 (4 eV), and E5 (5 eV) (for convenience you can call energy states as compartments/boxes), find total number of microstates corresponding to the condition “E0 box” contains 2 particles and total energy remains 3 eV. Also draw the schematic diagram showing distinguishable particles occupying the energy levels.Region 1 is x 0 with &2 = 460. If E2 = 6a - 10a, + 8a: V/m, (a) find P1, and P2, (b) calculate the energy densities in both regions.When two spiral galaxies collide, the stars generally do not run into each other, but the gas clouds do collide, triggering a burst of new star formation. (a) Estimate the probability that our Sun would collide with another star in the Andromeda Galaxy, if a collision between the Milky Way and Andromeda were happening at the present time. To simplify the problem, assume that each galaxy has 100 billion stars exactly like the Sun spread evenly over a circular disk with a radius of 100,000 light-years. (Hint: First calculate the total area of 100 billion circles with the radius of the Sun and then compare that total area to the area of the galactic disk.) (b) Estimate the probability that a gas cloud in our galaxy could collide with another gas cloud in the Andromeda Galaxy. To simplify the problem, assume that each galaxy contains 100,000 gas clouds of warm hydrogen gas, that each cloud has a radius of 300 light-years, and that these clouds are spread evenly over a circular disk with a…