1. Show that the kinetic energy of a three-dimensional gas of N free electrons at OK is U₁ = N EF 2. Pressure and bulk modulus of an electron gas (a). Derive a relation connecting the pressure and volume of an electron gas at 0 K. Hint: Use the result of problem 1 and the relation between EF and electron concentration(b) Show that the bulk modulus B --V(ap/av)of an electron gas at OK is B = 5p/3=1000/gy. (c) Estimate for potassium (see table 1), the value of electron gas contribution to B 3. Chemical potential in two dimensions. Show that the chemical potential of a Fermi gas in two dimensions is given by μ(T) = kaT In[exp(anh2/mkaT)-1]for n electrons per unit area. Note: The density of orbitals of a free electron gas in two dimension is independent of energy: D(e) m/h²m per unit area of specimen = 4. The number of electrons is, per unit volume, n = de D(e) where D(E)is the density of orbitals. In two dimensions n = m m de=(+ log(1+e¯/)) ΠΑΣ where the define integral is evaluated with the help of Dwight (569,1] 5. Fermi gases in astrophysics, (a) Given Me = 2x10 g for the mass of the Sun, estimate the number of electrons in the Sun. In a white dwarf star this number of electrons may be ionized and contained in a sphere of radius 2 X 10° cm; find the Fermi energy of the electrons in electron volts, (b) The energy of an electron in the relativistic limit e >> me² is related to the wavevector ase pc = hkc. Show that the Fermi energy in this limit is 6 hc (N/V) 1/3, roughly, (c) If the above number of electrons were contained within a pulsar of radius 10 km, show that the Fermi energy would be 108 eV. This value explains why pulsars are beheved to be composed largely of neutrons rather than of protons and electrons, for the energy release in the reaction n→ p+ e is only. 0.8 x 10° eV, which is not large enough to enable many electrons to form a Fermi sea. The neutron decay proceeds only until the electron concentration builds up enough to create a Fermi level of 0.8 X 10" eV, at which point the neutron, proton, and electron concentrations are in equilibrium. 6. The atom He' has spin % and is a fermion. The density of liquid He is 0.081 g cm³ near absolute zero. Calculate the Fermi energy Є, and the Fermi temperature T.