5. Given the single particle (ideal perfect gas) partition function Z = particle ideal gas equation. 2mL²kBT (2m²), derive the single
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- 4. The first excited state of the helium atom lies at an energy 19.82 eV above the ground state. If this excited state is three-fold degenerate while the ground state is non-degenerate, find the relative populations of the first exited and the ground states for helium gas in thermal equilibrium at 10,000 K.3. Ignoring spin of the Fermions in an ideal Fermi Gas, the maximal occupation number of an energy is: A) Infinite B) One E) Zero mk T C) Two 4. For a classical ideal gas trapped in a box of volume I, the thermal De-Broglie wavelength can be written as At In this case, the canonical partition function for a single particle of mass m is: A) 12² ) V2¹ D) Three D) E) Independent of Temperature2. An electron is confined to a nanowire 2 nm in length. Model this system as a 1-D particle-in-a-box, going from 0 to 2 nm. a. Compute the probability that the electron is in the range 0.95 ≤ x ≤ 1.05 nm for the states n = 1, 2, 3. b. Compute the probability that the electron is located within a distance of 0.05 nm of the left end of the wire for each of the states n = 1, 10, 100.
- 1. Use the I function to do the following: TZ (a) prove that z!(-z)! : where z is any real number sin(TZThe most probable thermal distribution for the states of a system of particles is defined by having the probability of the th state, P₁, proportional to... the number of particles, N Oits Boltzmann factor O volume O pressure O the number of moles, n O temperature ►In discussing molecular rotation, the quantum num- ber J is used rather than 7. Using the Boltzmann distribution, calculate nj/no for ¹H³5Cl for J = 0, 5, 10, and 20 at T = 1025 K. Does ny/no go through a maximum as J increases? If so, what can you say about the value of J corre- sponding to the maximum?
- 5. A microscopic oscillator has its first and second excited states 0.05 eV and 0.10 eV above the ground-state energy. Calculate the Boltzmann factor for the ground state, first excited state, and second excited state, at room temperature.8. If 4(x) = D sin 17x, what is the probability density for the range L 0 to L? Show calculations.A quantum system has a ground state with energy Eo = 0 meV and a 2-fold degenerate excited state with energy E₁ = 50 meV. E1 Calculate the probability of finding the system in its ground state when it is at T = 300 K. Select one: O a. 0.78 O b. 0.22 O c. 1 O d. 0.87