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Problem 6.15. Mean energy of a toy model of an ideal Bose gas (a) Calculate the...
1) Consider a uniform system of extremely relativistic (i.e., &p=cp) Bose gas with N particles in...
1) Consider a uniform system of extremely relativistic (i.e., &p=cp) Bose gas with N particles in three-dimensions. (a) Calculate the density of states using the formula D(e) - .86 - c). (b) Find the Bose-Einstein condensation temperature T.. (e) Find the fraction of condensed bosons No/N as a function of T/T. (d) Find the total energy (E) for T <T.
Problem One (10 points) An ideal Bose gas consists of particles in volume V at thermal...
Problem One (10 points) An ideal Bose gas consists of particles in volume V at thermal equilibrium. 1. Evaluate the transition temperature of the gas in term of N. V. and mass m of the particles. 2. Show that the heat capacity, C, of the gas at low temperature ( 7 72) is proportional with T3, whereas it is constant at high temperature,
Consider an 3-dimensional ideal bose gas system whose dispersion relation is given by a) Find the mean occupation numb...
Consider an 3-dimensional ideal bose gas system whose dispersion relation is given by a) Find the mean occupation number of quantum state with a wave vector b) Find the total number of particles at excited states and internal energy at temperature and express it in terms of Bose-Einstein integral and thermal wave length h2k2 E hw 2m We were unable to transcribe this imageWe were unable to transcribe this imageU (T We were unable to transcribe this imagegn(z; h2 1/2...
(80 pts) Entropy of the ideal gas: Consider a monatomic gas of identical non-interacting particles of...
(80 pts) Entropy of the ideal gas: Consider a monatomic gas of identical non-interacting particles of mass m. The kinetic energy of each particle is given in terms of its momentum p by Ekinp2/(2m). For a given total energy U, volume V and total particle number N (with N » 1) calculate the entropy S(U, V,N)-klog Ω(U, V,N), where Ω(U, V,N) counts the number of different microscopic configurations for given N, U and V. To get a finite number for...
(12%) Consider a system of non-interacting fermions in equilibrium with a heat bath at temperature T...
(12%) Consider a system of non-interacting fermions in equilibrium with a heat bath at temperature T and a particle reservoir at chemical potential fl. Assume that we can neglect different spin orientations of the fermions. Each particle can be in one of three single-partiele states with energies 0, A and 2A. (a) Find the grand partition function of the system. (b) Find the mean number of particles and mean energy of the system. (C) Find the most probable microstate of...
An ideal gas enclosed in a volume V is composed of N identical particles in equilibrium...
An ideal gas enclosed in a volume V is composed of N identical particles in equilibrium at temperature T. (a) Write down the N-particle classical partition function Z in terms of the single-particle partition function ζ, and show that Z it can be written as ln(Z)=N(ln (V/N) + 3/2ln(T)+σ (1) where σ does not depend on either N, T or V . (b) From Equation 1 derive the mean energy E, the equation of state of the ideal gas and...
Q.7) Consider a systems of N>>1 identical, distinguishable and independent particles that can be placed in...
Q.7) Consider a systems of N>>1 identical, distinguishable and independent particles that can be placed in three energy levels of energies 0, E and 2€, respectively. Only the level of energy sis degenerate, of degeneracy g=2. This system is in equilibrium with a heat reservoir at temperature T. a) Obtain the partition function of the system. b) What is the probability of finding each particle in each energy level? c) Calculate the average energy <B>, the specific heat at constant...
ror n mor ot an ideal gas in a process (T, V) → (I Calculate the...
ror n mor ot an ideal gas in a process (T, V) → (I Calculate the average energy (E) for three state (E1S e2 < E3) particles at T Discuss variation of (E) when τ→0. 13.
A crucial step in obtaining the Fermi-Dirac and Bose-Einstein statistic is the equivalence shown below: nmax...
A crucial step in obtaining the Fermi-Dirac and Bose-Einstein statistic is the equivalence shown below: nmax [la-»* = [TECH kno Convince yourself of this identity by showing it is valid in a case where each of 3 energy levels can host up to two particles, thus nk 0.1,2; k= 1, 2, 3. 1. Consider a gas on non-interacting magnetic molecules. Consider that when a magnetic field is applied, these molecules can align parallel or antiparallel to the field and the...
Problem C The partition function for an ideal gas is given by integrating over all possible...
Problem C The partition function for an ideal gas is given by integrating over all possible position and momen- tum configurations, weighted by a Boltzmann factor, for each particle (6 integrals per particle over z, v, z, pz, py, pz _ each running from-oo to +oo) and multiplying all N of these together (the factor of h is included to cancel the dimensions of dpdr; the factor of N! is included to divide out the multiplicity of particle-particle exchange) a)...
1) Consider a uniform system of extremely relativistic (i.e., &p=cp) Bose gas with N particles in three-dimensions. (a) Calculate the density of states using the formula D(e) - .86 - c). (b) Find the Bose-Einstein condensation temperature T.. (e) Find the fraction of condensed bosons No/N as a function of T/T. (d) Find the total energy (E) for T <T.
Problem One (10 points) An ideal Bose gas consists of particles in volume V at thermal equilibrium. 1. Evaluate the transition temperature of the gas in term of N. V. and mass m of the particles. 2. Show that the heat capacity, C, of the gas at low temperature ( 7 72) is proportional with T3, whereas it is constant at high temperature,
(80 pts) Entropy of the ideal gas: Consider a monatomic gas of identical non-interacting particles of mass m. The kinetic energy of each particle is given in terms of its momentum p by Ekinp2/(2m). For a given total energy U, volume V and total particle number N (with N » 1) calculate the entropy S(U, V,N)-klog Ω(U, V,N), where Ω(U, V,N) counts the number of different microscopic configurations for given N, U and V. To get a finite number for...
(12%) Consider a system of non-interacting fermions in equilibrium with a heat bath at temperature T and a particle reservoir at chemical potential fl. Assume that we can neglect different spin orientations of the fermions. Each particle can be in one of three single-partiele states with energies 0, A and 2A. (a) Find the grand partition function of the system. (b) Find the mean number of particles and mean energy of the system. (C) Find the most probable microstate of...
Q.7) Consider a systems of N>>1 identical, distinguishable and independent particles that can be placed in three energy levels of energies 0, E and 2€, respectively. Only the level of energy sis degenerate, of degeneracy g=2. This system is in equilibrium with a heat reservoir at temperature T. a) Obtain the partition function of the system. b) What is the probability of finding each particle in each energy level? c) Calculate the average energy <B>, the specific heat at constant...
ror n mor ot an ideal gas in a process (T, V) → (I Calculate the average energy (E) for three state (E1S e2 < E3) particles at T Discuss variation of (E) when τ→0. 13.
A crucial step in obtaining the Fermi-Dirac and Bose-Einstein statistic is the equivalence shown below: nmax [la-»* = [TECH kno Convince yourself of this identity by showing it is valid in a case where each of 3 energy levels can host up to two particles, thus nk 0.1,2; k= 1, 2, 3. 1. Consider a gas on non-interacting magnetic molecules. Consider that when a magnetic field is applied, these molecules can align parallel or antiparallel to the field and the...
Problem C The partition function for an ideal gas is given by integrating over all possible position and momen- tum configurations, weighted by a Boltzmann factor, for each particle (6 integrals per particle over z, v, z, pz, py, pz _ each running from-oo to +oo) and multiplying all N of these together (the factor of h is included to cancel the dimensions of dpdr; the factor of N! is included to divide out the multiplicity of particle-particle exchange) a)...
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