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1. 80 Consider a system where a particle can...
Consider a paramagnetic system of N elementary dipoles (with dipole moment μ) that can only have ...
Using MATLAB
Consider a paramagnetic system of N elementary dipoles (with dipole moment μ) that can only have states of parallel ↑ or antiparallel ↓ to the applied magnetic field B. The energies associated with each dipole is ±μ-B, the lower energy state being when the dipole is parallel to the B field. The macrostate state of the system will be defined by Nt, or equivalently, the total energy: The multiplicity of a given macrostate of a paramagnet is given...
Imagine a particle that can exist in only 3 states or levels. These energies levels are...
Imagine a particle that can exist in only 3 states or levels. These energies levels are –0.1 eV, 0.0 eV, and +0.1 eV. This particle is in eq with a reservoir at T = 300 K. A) Find the partition function for this particle. I want a number and a unit. Comment on what you would expect, and what the partition function tells you! B) Find the relative probability (%) for the particle to be in each of these...
Consider a system of distinguishable particles having only three energy levels (0, 1 and 2) equally...
Consider a system of distinguishable particles having only three energy levels (0, 1 and 2) equally separated by an energy , delta e, which is equal to the value of kT at 25 K. Calculate at 25 K: (a) the ratios of populations n1/n0 and n2/n0 (b) the molecular partition function, q (c) the molar internal energy, E = U - U(0), in J/mol (d) the molar entropy, S, in J/(K mol) (e) the molar constant volume heat capacity, Cv,...
A system has 11 energy levels with an equidistant spacing of delta E = 1 kJ...
A system has 11 energy levels with an equidistant spacing of delta E = 1 kJ mol-1. (a) In a single Excel graph, plot the population probabilities pi of all levels for the temperatures T = 3 K, 300 K, and 6000 K. (energy on the x-axis, probability on the y-axis) Note that: 3 K temperature of deep space 300 K “room temperature” 6000 K surface temperature of the sun. (b) Determine the value of the partition function...
Question 9 Consider a quantum system comprising two indistinguishable particles which can occupy only three individual-particle...
Question 9 Consider a quantum system comprising two indistinguishable particles which can occupy only three individual-particle energy levels, with energies 81 0, 82 2 and E3 38.The system is in thermal equilibrium at temperature T. (a) Suppose the particles which can occupy an energy level. are spinless, and there is no limit to the number of particles (i) How many states do you expect this system to have? Justify your answer (ii) Make a table showing, for each state of...
The partition function at constant V, T is the sum of Boltzmann factors where the sum is over independent states, . Week #10's Lesson showed that Z is related to two state functions, U and F F=-k...
The partition function at constant V, T is the sum of Boltzmann factors where the sum is over independent states, . Week #10's Lesson showed that Z is related to two state functions, U and F F=-kT lnZ Using these two relations, derive the relationships between Z and the following state functions and C (a) Entropy: S k In Z + kT (oln Z/oT)v (b) Pressure: P- kT (oln ZƠVr (d) Gibbs Free Energy: G-kTInZ + kTV(OlnZ/aV)T (e) Heat Capacity:...
1 The Gibbs Paradox Consider N particles, each of mass m, in a 3-dimensional volume V at temperature T. Each particle i...
1 The Gibbs Paradox Consider N particles, each of mass m, in a 3-dimensional volume V at temperature T. Each particle i has momentum pi. Assume that the particles are non-interacting (ideal gas) and distinguishable. a) (2P) Calculate the canonical partition function N P for the N-particle system. Make sure to work out the integral. b) (2P) Calculate the free energy F--kBTlnZ from the partition function Z. Is F an extensive quantity? c) (2P) Calculate the entropy S F/oT from...
Parts e, f, and g only please 2. Let f(x) = -3x + 2 for 0...
Parts e, f, and g only please
2. Let f(x) = -3x + 2 for 0 < x < 1. (a) If we partition the interval (0, 1) into five subintervals of equal length Ar, 0 = xo <12 <2<83 < 14 < 25 < x6 = 1, what is Ar and what are the ri? (b) Sketch a diagram for each of L5 and R5, the left and right enpoint Riemann sums for f(c) using the partition above. (c)...
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...
1. Consider a quantum system comprising three indistinguishable particles which can occupy only three individual-partic...
1. Consider a quantum system comprising three indistinguishable particles which can occupy only three individual-particle energy levels, with energies ε,-0, ε,-2e and ε,-3. The system is in thermal equilibrium at temperature T. Suppose the particles are bosons with integer spin. i) How many states do you expect this system to have? Justify your answer [2 marks] (ii) Make a table showing, for each state of this system, the energy of the state, the number of particles (M, M,, N) with...
Using MATLAB
Consider a paramagnetic system of N elementary dipoles (with dipole moment μ) that can only have states of parallel ↑ or antiparallel ↓ to the applied magnetic field B. The energies associated with each dipole is ±μ-B, the lower energy state being when the dipole is parallel to the B field. The macrostate state of the system will be defined by Nt, or equivalently, the total energy: The multiplicity of a given macrostate of a paramagnet is given...
Question 9 Consider a quantum system comprising two indistinguishable particles which can occupy only three individual-particle energy levels, with energies 81 0, 82 2 and E3 38.The system is in thermal equilibrium at temperature T. (a) Suppose the particles which can occupy an energy level. are spinless, and there is no limit to the number of particles (i) How many states do you expect this system to have? Justify your answer (ii) Make a table showing, for each state of...
The partition function at constant V, T is the sum of Boltzmann factors where the sum is over independent states, . Week #10's Lesson showed that Z is related to two state functions, U and F F=-kT lnZ Using these two relations, derive the relationships between Z and the following state functions and C (a) Entropy: S k In Z + kT (oln Z/oT)v (b) Pressure: P- kT (oln ZƠVr (d) Gibbs Free Energy: G-kTInZ + kTV(OlnZ/aV)T (e) Heat Capacity:...
1 The Gibbs Paradox Consider N particles, each of mass m, in a 3-dimensional volume V at temperature T. Each particle i has momentum pi. Assume that the particles are non-interacting (ideal gas) and distinguishable. a) (2P) Calculate the canonical partition function N P for the N-particle system. Make sure to work out the integral. b) (2P) Calculate the free energy F--kBTlnZ from the partition function Z. Is F an extensive quantity? c) (2P) Calculate the entropy S F/oT from...
Parts e, f, and g only please
2. Let f(x) = -3x + 2 for 0 < x < 1. (a) If we partition the interval (0, 1) into five subintervals of equal length Ar, 0 = xo <12 <2<83 < 14 < 25 < x6 = 1, what is Ar and what are the ri? (b) Sketch a diagram for each of L5 and R5, the left and right enpoint Riemann sums for f(c) using the partition above. (c)...
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...
1. Consider a quantum system comprising three indistinguishable particles which can occupy only three individual-particle energy levels, with energies ε,-0, ε,-2e and ε,-3. The system is in thermal equilibrium at temperature T. Suppose the particles are bosons with integer spin. i) How many states do you expect this system to have? Justify your answer [2 marks] (ii) Make a table showing, for each state of this system, the energy of the state, the number of particles (M, M,, N) with...
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