MSc Physics (Electronics I) SEM III 2022 2023 Jan 2023 PHYSICS STATISTICAL MECHANIC Question Paper - Mumbai University | munotes

Questions asked in this paper

• (2) Figures to the right indicate full marks

1. Q1 (a) Attempt any one:
   • (i) Show that the change in the entropy due to mixing of two ideal gases 8 results in to the paradox. Explain how the paradox is resolved
   • (ii) Consider two systems in contact with each other and isolated from 8 surrounding. Using statistical concept of possible microstates accessible for the composite system, show that at equilibrium:T, = T, , P, =P, and
   • (b) Attempt any one:
   • (i) the problem of one-dimensional simple harmonic oscillator. 4 Hence show that the fundamental volume of the phase space is given
   • (ii) Let the entropy of the classical ideal gas is given by 4 = Nk 42 ke . Obtain equation for th (N,V,E) = n 5 . ain equation for the specific heat at constant volume C, and the specific heat at constant
2. Q2 (a) Attempt any one:
   • (i) State and prove Equipartition theorem. 8
   • (ii) Consider a system of N quantum harmonic oscillators with frequency w. 8 Derive the expression for N-particle partition function and prove that the Helmholtz free energy for the system is
   • (b) Attempt any one:
   • (i) that the energy fluctuations AF in canonical ensemble follows: 4
   • (ii) that the N-particle partition function for a classical ideal gas is given 4
3. Q3 (a) Attempt any one:
   • (i) the relation for density fluctuation in grand canonical ensemble 8 where Kr is isothermal compressibility of the system
   • (ii) do you mean by phase equilibrium? Show that Gibb’s free energy is 8 minimized at equilibrium. Hence derive the Clausius-Clapeyron equation
   • (b) Attempt any one:
   • (i) Sketch P-T phase diagram for Helium-4. State its properties in different 4
   • (ii) function of a system of independent localized particles is given by 4 Obtain the expression for pressure P and number of particles N
4. Q4 (a) Attempt any one:
   • (i) harmonic oscillator, show that 8
   • (ii) Derive an expression for antisymmetric wave function of indistinguishable 8
   • (b) Attempt any one:
   • (i) a short note on wave functions of indistinguishable particles. 4
   • (ii) State and explain the postulate of equal a priori probabilities. 4
5. Q5 Attempt any four:
   • (a) Show that for micro-canonical ensemble: 3
   • (b) Explain the phase space of a classical system. Hence discuss the concept of 3
   • (c) Forasystem in canonical ensemble, show that C, = —T 3
   • (d) What is the ‘virial’ for a system? State virial theorem. 3
   • (e) Define fugacity z of the system. Show q-potential is logarithm of the grand 3
   • (f) Calculate the slope of the solid-liquid transition line for water near the triple 3 point T = 273.16K, given that the latent heat of melting is 80cal/g, the density of the liquid phase is and the density of the ice phase is Estimate the melting temperature at P = 100atm
   • (g) What are the Hamiltonian and the wavefunction of a free particle in a three 3 dimensional box of length L?
   • (h) Write an expression for mean thermal wavelength. Explain each term. 3

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