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The pressure for a noninteracting Fermi gas with internal energy U at temperature T is given as . Find value of x?
The mean pressure of the perfect Fermi gas can be written as
The average energy of the system is
For an ideal fermion is a cubical box, the energy of single particle states are given by
If there is no interaction
E_{i} = energy of i^{th} microstate of the system.
The correct answer is: 0.667
During compression of a spring, the work done is 10 kJ and 2 kJ escaped to the surrounding as heat. The change in internal energy ∆U (in kJ) is
w = 10 kJ
q = –2 kJ
According to the first law of thermodynamics, ∆U = q + w
= –2 + 10 = 8 kJ
According to FermiDirac statistic the number of particles in a phase cell is.
Systems following FermiDirac statistics obey Pauli’s exclusion principal and each phase cell corresponding to one energy level which can be occupied by only 1 particle.
The correct answer is: 1
Six distinguishable particles are distributed over three nondegenerate levels of energies 0, E and 2E. The total energy of the distribution for which the probability is a maximum is (in terms of E)?
Since, the levels are nondegenerate there is only one state associated with each energy.
Let the number of particles is 3 energy state be N_{1} , N_{2} and N_{3} respectively.
where N_{1} + N_{2} + N_{3} = 6
As the particles are distinguishable, the number of microstates i.e. the number of ways of choosing N_{1} , N_{2} and N_{3} particles from 6 particles is
W → thermodynamic probability. It is maximum when N_{1} !N_{2} !N_{3} ! is a minimum
where N_{1} = N_{2} = N_{3} = 2
The corresponding total energy of distribution is
= 6E
The correct answer is: 6
Consider a radiation cavity of volume V at temperature T. The average number of photons in equilibrium is proportional to T^{α} , Find α?
For photons, BE statistics is applicable
For photons,
the density of states for photons between p and p + dp is
Since two independent direction of polarization
hv = dE
⇒ hdv = dE
Average number of photons
The correct answer is: 3
Ratio is average energy of an electron in a metal at T = 0 to the Fermi energy at T = 0 is.
The total number of cells upto energy level
Thus, the number of cells corresponding to energy interval E and E + dE is given by
The total number of cells between energy level 0 and E_{F} at T = 0, are equal to the total number of particles i.e.
Thus,
E_{fo }is value of E_{F} at T = 0.
The average kinetic energy of the particle at T = 0 is given by
The correct answer is: 0.6
Consider a system of 2 identical particles each of which can be in any one of the 3 single particles states. The number states of the systems are possible in BE statistics?
The total number of way W, of distributing 2 identical particles in the given system is
n_{i} = 2, g_{i} = 2
The correct answer is: 6
The probability that a state which is 0.2eV above the Fermi energy in metal atom at 700K is.
Probability of occupancy of a given state with energy E is given by the FermiDirac distribution formula which is
f(E) = 0.035
The correct answer is: 0.035
The Fermi energy of a free electron gas depends on the electron density ρ as Find the value of α?
The Fermi energy of the electron gas is given by
with g_{s} = 2
The correct answer is: 0.667
In a system of particles, each particles can be in any one of three possible quantum states. The ratio of the probability that the two particles occupy the same state to the probability that the two particle occupy different state for BE statistics is.
For BE statististic, the particles are indistinguishable
So,
The correct answer is: 1
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