Suppose there is a flat, absorbing surface in the presence of a vapour of atoms. Each...
Suppose there is a flat, absorbing surface in the presence of a vapour of atoms. Each atom in the vapour can either be stuck to the surface with energy ε or remain unstuck. Assume that there are M sites on the surface at which the vapour atoms can stick. If N atoms are stuck at any moment, the total energy of that state would be Nε. Can you calculate the grand partition function for this system?
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Suppose there is a flat, absorbing surface in the presence of a
vapour of atoms. Each...
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...
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...
2. A gas that consists of N atoms, each of which and be in any of...
2. A gas that consists of N atoms, each of which and be in any of 3 levels, a, b or c, with energy levels Ea = -e, Ep = 0, Ec = +€ {+t E:0 Ea = -6 - (lna heat bait at constant I) a) Write down the 1 particle partition function 21, for (i) distinguishable particles and (ii) identical quantum particles (2.5 points) b) What are the probabilities for an atom to be in states a, b...
A. Suppose CO adsorbs onto a solid surface such that the CO bond is perpendicular to...
A. Suppose CO adsorbs onto a solid surface such that the CO bond is perpendicular to the surface, the oxygen atom is immobilized on the surface but the carbon atom can still move as the bond vibrates. Calculate the reduced mass for this surface-adsorbed CO molecule B. Calculate the vibration temperature θvib for this surface-adsorbed 12C16O molecule. Assume the CO bond force constant (k= 1860 Nm^-1) does not change when CO adsorbs to teh surface. C. Calculate the vibrational partition...
Dimoglobin as a case study in cooperativity. Note that each binding site can be occupied, ,...
Dimoglobin as a case study in cooperativity.
Note that each binding site can be occupied,
, or unoccupied,
, and that a parameter J describes the cooperativity between the O2
molecules when both states are occupied (i.e., the energy is not
just the sum of the individual binding energies).
a) Write down the weights (Gibbs factors) for each of the
different states for the dimoglobin system shown in Figure 2.
b) What is the formula that describes the energy of...
1. Consider a mass M moving near a flat surface (which we may take to be 0 in the presence of the...
1. Consider a mass M moving near a flat surface (which we may take to be 0 in the presence of the gravitational acceleration g 9.8 m/s2. (a) Show using the Wilson Sommerfeld Quantization rule that the amplitude of bounces To and the system energy are quantized. For this purpose, use: / pio It may be useful to review the example of the harmonic oscillator where we used p md/dt and q-r. In the case of this question, one full...
2. Interacting Spins (5 points each part, 30 points total). Two spins, each of which can...
2. Interacting Spins (5 points each part, 30 points total). Two spins, each of which can be in one of two states, up or down, are in equilibrium with a heat reservoir at temperature t. They interact as follows: When the two spins point in the same direction, their interaction energy is – J, and when they point in opposite directions, their interaction energy is J. The spins also each have a magnetic moment m and are subject to a...
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...
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...
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...
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...
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...
2. A gas that consists of N atoms, each of which and be in any of 3 levels, a, b or c, with energy levels Ea = -e, Ep = 0, Ec = +€ {+t E:0 Ea = -6 - (lna heat bait at constant I) a) Write down the 1 particle partition function 21, for (i) distinguishable particles and (ii) identical quantum particles (2.5 points) b) What are the probabilities for an atom to be in states a, b...
Dimoglobin as a case study in cooperativity.
Note that each binding site can be occupied,
, or unoccupied,
, and that a parameter J describes the cooperativity between the O2
molecules when both states are occupied (i.e., the energy is not
just the sum of the individual binding energies).
a) Write down the weights (Gibbs factors) for each of the
different states for the dimoglobin system shown in Figure 2.
b) What is the formula that describes the energy of...
1. Consider a mass M moving near a flat surface (which we may take to be 0 in the presence of the gravitational acceleration g 9.8 m/s2. (a) Show using the Wilson Sommerfeld Quantization rule that the amplitude of bounces To and the system energy are quantized. For this purpose, use: / pio It may be useful to review the example of the harmonic oscillator where we used p md/dt and q-r. In the case of this question, one full...
2. Interacting Spins (5 points each part, 30 points total). Two spins, each of which can be in one of two states, up or down, are in equilibrium with a heat reservoir at temperature t. They interact as follows: When the two spins point in the same direction, their interaction energy is – J, and when they point in opposite directions, their interaction energy is J. The spins also each have a magnetic moment m and are subject to a...
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...
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...
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...
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