A proton inside a nucleus
may have speed upto
. How many quantum states are available to it?
We are asked to find the number of quantum states available to a proton inside a nucleus. We will solve this numerically using the particle-in-a-box model in 3D. Given data: * Mass of proton: ( m = 1.67 \times 10^{-27} , \text{kg} ) * Radius of nucleus: ( R = 10^{-14} , \text{m} ) * Maximum speed: ( v = 2.0 \times 10^8 , \text{m/s} ) * Planck’s constant: ( h = 6.626 \times 10^{-34} , \text{J·s} ) --- Step 1: ________ ______ _____ ______ ____.
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Using the relation of thermodynamic probability
Obtain an expression of Fermi Dirac distribution function. Show a plot of (the occupation index of a state corresponding to energy
) versus
for different temperatures and discuss it. What is the physical interpretation of Fermi energy?
A proton inside a nucleus
may have speed upto
. How many quantum states are available to it?
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.
Show under what condition Poisson distribution tends to Normal distribution.
What is the Bose-Einstein condensation? Write an expression of number of particles in the excited state (Nex) in terms of Bose-Einstein condensation temperature (Tc), total number of bosons (N) in an assembly and temperature T. Show a plot of distribution of bosons as a function of temperature below and above the Tc.
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.