QUESTION 3 [TOTAL MARKS: 25] The Fermi energy ε of metallic sodium is 3.23 eV. Q3(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 describe sodium at 273 K. Use appropriate diagrams in your answer showing the distribution of the electrons at a temperature 7 of 0 K and 273 K, respectively, as a function of their energy. Q3(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 Ear of the free electron gas in sodium at 300 K. av Q3(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. The density of states g(a) of the free electron gas is given by g(ε)=- 8лmV (2mε). h³ The Fermi-Dirac distribution function f(ε) can be written in the following form 1 f(ε)= E-EF 1+eksT