QUESTION 5 [TOTAL MARKS: 25] The Fermi energy, of metallic sodium is 3.23 eV. Q5(a) [6 marks] Calculate the Fermi temperature T, of sodium and, hence, justify the use of the degenerate (ie at 0 K) electron gas model to sodium at 273 K. Use appropriate diagrams in your answer showing the distribution of the electrons at a temperature T of 0 K and 273 K, respectively, as a function of their energy. Q5(b) [11 marks] Show that the total number of electrons contained in a volume V of sodium at 0 K can be written as an integral of the density of states. Use this result to estimate the number density and the average energy E of the free electron gas in sodium at 300 K. Q5(c) [8 marks] Obtain an expression for the pressure P of the free electron gas as a function of its Fermi energy and number density if P = (2E/3V) where E is the total internal energy of the gas and V is the volume. Estimate P for metallic sodium at 300 K. 8mV h³ The density of states g(e) of the free electron gas is given by g(e)=- (2me) The Fermi-Dirac distribution function f(e) can be written in the following form 1 f(e)=- 8-8 1+e