Chemical potential in two dimensions. Show that the chemical potential of aFermi gas in two dimensions is given by:M(T) = kgT In [exp(mnh/mk;T)– 1] ,for n electrons per unit area. Note: The density of orbitals of a free electron gas intwo dimensions is independent of energy: D(e) = m/Th", per unit area of specimen.

Answered: Image /qna-images/answer/7c82acdf-71c9-41f5-a0db-6a16a4b51781.jpg

Chemical potential in two dimensions. Show that the chemical potential of a Fermi gas in two dimensions is given by: u(T) = kgT ln [exp(mnh/mkgT) – 1], for n electrons per unit area. Note: The density of orbitals of a free electron gas in two dimensions is independent of energy: D(e) = m/Th", per unit area of specimen.

Chemical potential in two dimensions. Show that the chemical potential of a Fermi gas in two dimensions is given by: u(T) = kgT ln [exp(mnh/mkgT) – 1], for n electrons per unit area. Note: The density of orbitals of a free electron gas in two dimensions is independent of energy: D(e) = m/Th", per unit area of specimen.

Chemistry

10th Edition

ISBN:9781305957404

Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Chapter1: Chemical Foundations

Section: Chapter Questions

Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...

See similar textbooks

Similar questions

(b) A system with chemical potential u = 12.5 meV contains identical particles each with integer spin and has a single quantum state at energy E, = 0 meV. The temperature of the system is fixed at 290 K. To three significant figures, what is the ratio Pg/P1, where P is the probability of finding a single particle in the state Eo, and P3 is the probability of finding 3 particles in this state? P3/P (Input your answer as a number.)
(4) Electrons are no longer ejected from a sample of metal whenever it is illuminated by light with wavelength greater than 344nm. (a) Based on the table below, what is the most likely metal being illuminated? (b) If the sample is illuminated by light with wavelength 48.3nm, what is the stopping potential? (c) In the event this sample of metal were replaced by a metal with a smaller work function, but still illuminated with light with 48.3nm, would the kinetic energy of the ejected electrons increase or decrease? (d) What is the threshold frequency for cesium? Atom W (eV) Cesium 2.1 Calcium 2.9 Uranium 3.6 Cadmium 4.07 Lead 4.14 Niobium 4.3 Iron 4.5 Copper Cobalt 4.7 5 Gold 5.11 Platinum 6.35
Estimate the wave l ength of electrons that have been accelerat ed from rest through a potential difference of 1.00 kV.
Calculate the energy separations in joules, kilojoules per mole, and electronvolts, and as a wavenumber, between the levels (a) n = 3 and n = 1, (b) n = 7 and n = 6 of an electron in a one-dimensional nanoparticle modelled by a box of length 1.5 nm.
We can use the classical harmonic oscillator to think about molecular bonds. The HCI molecule has a force constant k = 481 N/m. For the mass, use the reduced mass, which is defined as µ = (m₁m₂)/(m₁+m₂). a) Plot the potential energy of HCl from -1 to 1 Å. What happens to the curvature of the potential as the force constant is varied? What does this mean physically? b) Plot position as a function of time for a total energy of 6 x 10-20 J. What is the period of the motion? How does the period change as the force constant is varied? Explain why this makes sense physically.
(c) The kinetic energy, K₁, of electrons emitted from a metal surface after irradiation with UV light of wavelength λ is given by: hc Kg = -4 where his Planck's constant (6.626 x 10-4 Js), c is the speed of light in a vacuum (2.99 x 108 m s¹), and is the work function of the metal surface. In a specific experiment, light with a wavelength of 266 nm was used to irradiate a cadmium (Cd) metal surface.
8.) Based on the reactions: :1.) 211,S + 30 2SOe + 2H,0) AH, =-1035.94 kJ 2SO + O) AH, = +197.78 k.J b.) 2S0 ) c.) 21,S + O 2S, + 2H,O AH, = +110.02 kJ Use Hesses Law to calculate AH for: 2S, + 30e)
(5) A sample of sodium phosphate is dissolved in water to give 100.00 mL of solution (Solution A). Then this solution is diluted 5 times to give the final solution (Solution B). Solution B is Known to contain 3.920 x 1022 number of PO43- ions, which has an absorption peak at 460 nm and absorbance value of 0.355 in a 1.000 cm cuvette. (i) Calculate the molar absorptivity of Solution B. (ii) What mass of sodium phosphate (MM 163.94 g/mol) dissolved in Solution A.
E7B.3(b) Which of the following functions can be normalized (in all cases the range for x is from x = -∞ to ∞, and a is a positive constant): (i) sin(ax); (ii) cos(ax) ex²? Which of these functions are acceptable as wavefunctions?
What is the kinetic energy of an electron that has been accelerated through a potential difference of 10.0 kV?
16.2 Consider 1.00 x 1022 4He atoms in a box of dimensions 1.0 cm x 1.0 cm x 1.0 cm. Calculate the occupancy of the first excited level at 1.0 mK, 2.0 K, and 4.0 K. Do the same for ³He. What conclusions might you draw from the results of your calculations?
(c) The kinetic energy, KE, of electrons emitted from a metal surface after irradiation with UV light of wavelength A is given by: hc Kε = 7-9 where h is Planck's constant (6.626 x 10-4 Js), c is the speed of light in a vacuum (2.99 x 108 m s¹), and is the work function of the metal surface. In a specific experiment, light with a wavelength of 266 nm was used to irradiate a cadmium (Cd) metal surface. Calculate the photon energy of the light used in the experiment, in Joules. (1) ₂4