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COLLEGE: SCIENCES   DEPARTMENT: MATHEMATICAL AND PHYSICAL SCIENCES

COURSE CODE: PHY 408    COURSE TITLE: STATISTICAL AND THERMAL PHYSICS

TIME 2 HRS 15 MINUTES     ANSWER FOUR QUESTIONS   EXAM 2019/2020

  Useful constants N_A = 6.02 X 10²³, K=8.6178x10^-5eV/k , h =6.626 x10^-34 JS, K =1.3805 x 10^-23J/k,

1. (a) (i) State the First law of thermodynamics

              (ii)  State two postulates of equilibrium of isolated system.

                    (b)   Define the following terms (i) Temperature (ii) Thermal equilibrium

        (c) (i) Compare and contrast the 3 differences or 3 similarities between Fermi-Dirac distribution and Bose-Einstein distribution.

          (ii) A gas occupies a volume of 0.30m³ exerting a pressure p = 2 x10⁵ N/m² at constant    pressure the volume expands to 0.45m³. Find the work done by the gas .When a process occurs at constant pressure it is called isobaric.

  2. (a) (i) State the 2^nd law of thermodynamics

              (b) Given the energy of the system as   E = (1/z){E_re^-BEr

                  Show that (i) E^/= KT² (d/dT)(lnz) (ii) if     p = -dE/dV, show that p = (1/B)(d lnZ/dV)

                    (iii) S = K lnz + (U/T)  (iv) F = -KT(ln z) (v) U= -kT (d/dN)(lnz)

       (c) (i)  Define the term entropy?

3(a) Define the following terms (i) Microscopic state (ii) Macroscopic state

(iii) Show that for mechanical equilibrium P₁/T₁ = P₂/T_2., (iv) for thermal equilibrium 1/T₁ = 1/T₂.

 (b) Define the following terms (i) Ensemble (ii) Micro canonical ensemble (iii) Partition function          (iv)  Statistical weight.

           (c) (I) State the 3^rd law of thermodynamics (II) Zeroth Law of Thermodynamics

     4 (a) The chemical potential of an ideal boson gas at a temperature of 5 T_B is -5.2 K T_B .If the energy Eigen value of a single particle state at this temperature is 4.0 K T_B. Determine the mean occupation number of bosons for the state. i.e.  Bose-Einstein distribution function

           (b) In a quantum mechanical ensemble given the number of molecule with velocity V as

                  N(v)=(4N /II^1/2 ) ((M)^1/2/ (2kT)^3/2 ) (v)² e^(-mv2/2KT)   

                 Obtain (i) That most probable velocity, Vp= (2KT/m)^1/2

                            (ii) That average velocity,  V_ave = (8KT/IIm) ^1/2

                         (iii) That root mean square velocity, V_rms = (3KT/m) ^1/2

   5(a) (i)  What are identical particles? (ii) Give two examples of bosons, and two examples of fermions

  (ii) Write the expression for (i) Fermi-Dirac distribution (ii) Bose Einstein distribution (iii)  Maxwell Boltzmann distribution.

                  (b) Calculate the mean free path, collision frequency, collision time and diameter of nitrogen   molecule at S.T.P given that  the coefficient of viscosity (ῃ) =16.6x 10^—6 Ns/m. The density (ℓ) of nitrogen  is 1.25Kg/m³, average speed of molecule (v) = 450m/s and the molecular density (n) = 2.7x10²⁵ molecule/m³ for Nitrogen at S.T.P.

       (c) What is the total possible state for each of the following?

          (i) Maxwell Boltzmann statistics with distinguishable particles?

           (ii) Bose-Einstein statistics with indistinguishable particle?

  6  (a) The chemical potential of an ideal fermions gas at a temperature of 4 T_B is 4.4 K T_B .If the energy Eigen value of a single particle state at this temperature is 7.0 K T_B. Determine the mean occupation number of fermions for the state. i.e.  Fermi-Dirac distribution function.

      (b) (i)  What is the total; possible state for each of the Fermi-Dirac statistics with indistinguishable particle

  (ii) Statistical dynamics deals with issue of microscopically modelling the speed of irreversible processes that are driven by     imbalances. Give three examples?

     (c) (i) What is statistical mechanics?

          (ii) State two concepts of Mechanics (in classical and quantum terms     

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