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.
In the canonical ensemble, a system is in thermal equilibrium with a heat reservoir at a fixed temperature (T). The probability (P_i) of the system being in a microstate (i) with energy (E_i) is given by the Boltzmann distribution: [ P_i = \frac{e^{-E_i/k_B T}}{Z} ] where (Z = \sum_i e^{-E_i/k_B T}) is the canonical partition function and (k_B) is the Boltzmann constant. In this ensemble, thermodynamic quantities fluctuate because the system can _______ __________ ______ ___ _____ _____ ___.
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Using entropy as a function of temperature and pressure, obtain the first Ehrenfest’s equation.
The time evolution of the quantum state, is given by the Schrödinger wave 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.
State the Virial theorem and prove Boyle’s law using it.
For the grand canonical ensemble, obtain an expression for the ensemble average energy and ensemble average particle numbers.
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.
What is meant by Cluster Integrals? Express B2 and B3 in terms of Cluster Integrals.