Using entropy as a function of temperature and pressure, obtain the first Ehrenfest’s equation.
The entropy (S) can be expressed as a function of temperature (T) and pressure (P), i.e., (S = S(T, P)). The differential of entropy is: [ dS = \left(\frac{\partial S}{\partial T}\right)_P dT + \left(\frac{\partial S}{\partial P}\right)_T dP ] From thermodynamics, (\left(\frac{\partial _______ ___ ___ ______ _______ _________ ________ _____ ___.
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State and prove quantum Liouville equation.
Discuss the concept of thermodynamic fluctuations in the canonical ensemble. Define energy fluctuations and derive an expression for the mean square fluctuation of energy. Using a classical ideal gas, show that the relative fluctuation in energy varies as and hence, justify why thermodynamic calculations are valid for ordinary macroscopic systems.
For the grand canonical ensemble, obtain an expression for the ensemble average energy and ensemble average particle numbers.
Show that the probability that a state with energy above the Fermi level
filled is equal to the probability of a state with energy
below the Fermi level
is empty.
N particles obey Maxwell-Boltzmann distribution. They are distributed among three states with energies and
. If the equilibrium energy of the system is approximately 3000kBT, calculate the total number of particles.
Using entropy as a function of temperature and pressure, obtain the first Ehrenfest’s equation.
The Hamiltonian for the non-relativistic, non-interacting, monoatomic ideal gas is given by
where m is the mass of the particle and U(qi) is the potential energy. Obtain an expression of the phase space volume of the energy shell and hence obtain an expression of number of microstates.