Problem 7: Consider a system on N localized non interacting paramagnetic ions of spin- ½ and magnetic moment μ in an external magnetic field H is in thermal equilibrium at temperature T. Obtain the canonical partition function Z . Calculate magnetization M and susceptibility χ. Check that at high temperature, χ is inversely proportional to T .

Problem 8 : Consider a monatomic ideal gas of N particles enclosed in a volume V . The system is in thermodynamic equilibrium with a heat bath at temperature T and at constant chemical potential μ. Calculate the grand canonical partition function Q , obtain the grand potential . Verify the equation of state PV=Nk_B T .

Problem 9 : Consider N localized one dimensional quantum Harmonic oscillators of frequency ω in thermodynamic equilibrium at temperature T and chemical potential μ . Obtain the grand canonical partition function .

Problem 10: Consider a single component system of volume V, having two phases - solid and vapour, in equilibrium at temperature T. Treating the vapour as a monatomic ideal gas and the solid as quantum harmonic oscillator, show that a solid phase exists below a characteristic temperature given by where N is the total number of particles in the system, and .

[ These problems are usual text book problems and can be found as examples in the text books given in the references.]




References

[1] S. R. A. Salinas, Introduction to Statistical Physics , (Springer, New York, 2004).

[2] R. K. Pathria, Statistical Mechanics , (Butterworth-Heinemann, Oxford, 1996).

[3] K. Huang, Statistical Mechanics , (John Wiley & sons, New York, 2000).

[4] M. Plischke and B. Bergersen, Equilibrium Statistical Physics , (World Scientific, Singapore, 1994).

[5] J. K. Bhattacharjee, Statistical Physics: Equilibrium, and Non-equilibrium Aspects , (Allied Publishers Ltd., New Delhi, 1997).

[6] S. B. Santra and P. Ray, Statistical Mechanics and Critical Phenomena: A brief overview , in Computational Statistical Physics, edited by S. B. Santra and P. Ray, (Hindustan Book Agency, New Delhi, 2011).

[7] R. Kubo, Statistical mechanics: an advanced course with problems and solutions (North Holland, Amsterdam, 1965).