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Problem C The partition function for an ideal gas is given by integrating over all possible...
3. It can be shown that the canonical partition function of an N-particle monatomic ideal gas con...
Please be specific about the solution and thank you so much!
3. It can be shown that the canonical partition function of an N-particle monatomic ideal gas confined to a container of volume V at temperature T is given by 3 Use this partition function to derive an expression for the average energy and the constant- volume heat capacity of the monatomic ideal gas. Note that in classical thermodynamics these quantities were simply given. Your calculations show that these quantities...
Problem 3: (40 points) One-dimensional relativistic gas: Here we consider a non-interacting gas of N relativistic...
Problem 3: (40 points) One-dimensional relativistic gas: Here we consider a non-interacting gas of N relativistic particles in one dimension. The gas is confined in a container of length L, i.e., the coordinate of each particle is limited to 0 <q < L. The energy of the ith particle is given by ε = c (a) Calculate the single particle partition function Z(T,L) for given energy E and particle number N. [12 points] (b) Calculate the average energy E and...
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...
Answer a total of any THREE out of the four questions. Put the solution to each problem in a separate blue book and put the number of the problem and your name on the front of each book. If you su...
Answer a total of any THREE out of the four questions. Put the solution to each problem in a separate blue book and put the number of the problem and your name on the front of each book. If you submit solutions to more than three problems, only the first three problems as listed on the exam will be graded. Some possibly useful information: Sterling's asymptotic series: N In N-N + 1 ln(2nN] In N! N → oo, as 2...
Statistical_Mechanics 2 20 points) 2D ideal Fermi gas 24 Consider an ideal Fermi gas in 2D....
Statistical_Mechanics
2 20 points) 2D ideal Fermi gas 24 Consider an ideal Fermi gas in 2D. It is contained in an area of dimensions L x L. The particle mass is m. (a) Find the density of states D(e) N/L2 (b) Find the Fermi energy as a function of the particle density n = (c) Find the total energy as a function of the Fermi energy ef. (d) Find the chemical potential u as a function of n and T....
1. Show that for a classical ideal gas, Q1 alnQ1 NK Hint: Start with the partition function for t...
1. Show that for a classical ideal gas, Q1 alnQ1 NK Hint: Start with the partition function for the classical ideal gas ( Q1) and use above equation to find the value of right-hand side and compare with the value of r we derive in the class. (Recall entropy you derived for classical gas) NK Making use of the fact that the Helmholtz free energy A (N, V, T) of a thermodynamic system is an extensive property of the system....
Problem 4 A monoatomic Boltzmann ideal gas if spin / atoms in uniform magnetic field (B),...
Problem 4 A monoatomic Boltzmann ideal gas if spin / atoms in uniform magnetic field (B), has in additional to its usual kinetic translational energy, a magnetic energy of tuB per atom, where is u the magnetic moment. It is assumed that the gas is so diluted that the interaction of magnetic moments can be neglected. a) What is the partition function for a canonical ensemble of N such atoms. b) Calculate C, from the partition function. c) Draw the...
Find the partition function of a gas of N identical molecules of mass m contained in a vertical cylinder of height L and radius R placed in a uniform gravitational field. a. b. Demonstrate that for L...
Find the partition function of a gas of N identical molecules of mass m contained in a vertical cylinder of height L and radius R placed in a uniform gravitational field. a. b. Demonstrate that for L-oo the internal energy becomes U c. Explain why this value is larger than for the ideal gas. NkT
Find the partition function of a gas of N identical molecules of mass m contained in a vertical cylinder of height L and radius R...
Pls show full working thank you Problem 4.1 Ideal gas equation of state from the Grand...
Pls
show full working thank you
Problem 4.1 Ideal gas equation of state from the Grand potential The Grand Canonical ensemble can make some calculations particularly simple. To derive the ideal gas equation of state, we first note that the canonical partition function of a set of N identical and indistinguishable particles is given by Z-z/N! , where z is the single particle partition function in the canonical ensemble a) Show that the Grand Canonical partition function is -žte®)" b...
1. Quick Exercises (a) In lecture 15, we showed that the canonical partition function, Q, is related to the Helmhol...
1. Quick Exercises (a) In lecture 15, we showed that the canonical partition function, Q, is related to the Helmholz free energy: A = -kTinQ. Using the fundamental thermo- dynamic relation of Helmheltz free energy (i.e. dA = -SIT - PdV + pdN), express P, and u in terms of Q. (b) The canonical partition function for N non-interacting, indistinguishable parti- cles in volume V at temperature T is given by Q(N, V,T) = where where 9(V, T) is the...
Please be specific about the solution and thank you so much!
3. It can be shown that the canonical partition function of an N-particle monatomic ideal gas confined to a container of volume V at temperature T is given by 3 Use this partition function to derive an expression for the average energy and the constant- volume heat capacity of the monatomic ideal gas. Note that in classical thermodynamics these quantities were simply given. Your calculations show that these quantities...
Problem 3: (40 points) One-dimensional relativistic gas: Here we consider a non-interacting gas of N relativistic particles in one dimension. The gas is confined in a container of length L, i.e., the coordinate of each particle is limited to 0 <q < L. The energy of the ith particle is given by ε = c (a) Calculate the single particle partition function Z(T,L) for given energy E and particle number N. [12 points] (b) Calculate the average energy E and...
Answer a total of any THREE out of the four questions. Put the solution to each problem in a separate blue book and put the number of the problem and your name on the front of each book. If you submit solutions to more than three problems, only the first three problems as listed on the exam will be graded. Some possibly useful information: Sterling's asymptotic series: N In N-N + 1 ln(2nN] In N! N → oo, as 2...
Statistical_Mechanics
2 20 points) 2D ideal Fermi gas 24 Consider an ideal Fermi gas in 2D. It is contained in an area of dimensions L x L. The particle mass is m. (a) Find the density of states D(e) N/L2 (b) Find the Fermi energy as a function of the particle density n = (c) Find the total energy as a function of the Fermi energy ef. (d) Find the chemical potential u as a function of n and T....
1. Show that for a classical ideal gas, Q1 alnQ1 NK Hint: Start with the partition function for the classical ideal gas ( Q1) and use above equation to find the value of right-hand side and compare with the value of r we derive in the class. (Recall entropy you derived for classical gas) NK Making use of the fact that the Helmholtz free energy A (N, V, T) of a thermodynamic system is an extensive property of the system....
Problem 4 A monoatomic Boltzmann ideal gas if spin / atoms in uniform magnetic field (B), has in additional to its usual kinetic translational energy, a magnetic energy of tuB per atom, where is u the magnetic moment. It is assumed that the gas is so diluted that the interaction of magnetic moments can be neglected. a) What is the partition function for a canonical ensemble of N such atoms. b) Calculate C, from the partition function. c) Draw the...
Find the partition function of a gas of N identical molecules of mass m contained in a vertical cylinder of height L and radius R placed in a uniform gravitational field. a. b. Demonstrate that for L-oo the internal energy becomes U c. Explain why this value is larger than for the ideal gas. NkT
Find the partition function of a gas of N identical molecules of mass m contained in a vertical cylinder of height L and radius R...
Pls
show full working thank you
Problem 4.1 Ideal gas equation of state from the Grand potential The Grand Canonical ensemble can make some calculations particularly simple. To derive the ideal gas equation of state, we first note that the canonical partition function of a set of N identical and indistinguishable particles is given by Z-z/N! , where z is the single particle partition function in the canonical ensemble a) Show that the Grand Canonical partition function is -žte®)" b...
1. Quick Exercises (a) In lecture 15, we showed that the canonical partition function, Q, is related to the Helmholz free energy: A = -kTinQ. Using the fundamental thermo- dynamic relation of Helmheltz free energy (i.e. dA = -SIT - PdV + pdN), express P, and u in terms of Q. (b) The canonical partition function for N non-interacting, indistinguishable parti- cles in volume V at temperature T is given by Q(N, V,T) = where where 9(V, T) is the...
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