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A1.2. In the lectures we have seen that the multiplicity of an Einstein solid with q...
Calcium carbonate can decompose into calcium oxide and carbon dioxide gas with the (c) reaction: CaCO3...
Calcium carbonate can decompose into calcium oxide and carbon dioxide gas with the (c) reaction: CaCO3 СаО + СО2 The change in enthalpy for this reaction is AH = +178 kJ/mol and the change in entropy is AS = +0.16 kJ/(K. mol) (i) Describe the concept of the Gibbs energy and how it is related to the enthalpy and entropy (ii Hence calculate the minimum temperature for the decomposition of calcium carbonate to proceed spontaneously An Einstein solid consists of...
[20 pts] It can be shown that the multiplicity of the macrostate with r quanta of...
[20 pts] It can be shown that the multiplicity of the macrostate with r quanta of energy in an Einstein solid with n oscillators is (a) For an Einstein solid with each of the following values of n and r, list all of the possible microstates, count them, and verify formula E. 1 above. (a-1) n 3,r 5 (a-2) n 3,r-7 (a-3) n 4, r 4 (a-4) n 1,r anything (a-5) n anything,1 (b) Compute the multiplicity of an Einstein...
T2B.2 Calculate the multiplicity of an Einstein solid with N = 1 and U = 5€...
T2B.2 Calculate the multiplicity of an Einstein solid with N = 1 and U = 5€ by directly listing and counting the microstates. Check your work by using equation T2.7. macrostate of an Einstein solid. R2(0,N) - Vece +3N-1)! (9)! (3N-1)! Write down all of the microstates for a hypothetian HALI 3 Is the
Question 4: (i) Write down the form of the Bose-Einstein distribution and discuss the phenomenon ...
Question 4: (i) Write down the form of the Bose-Einstein distribution and discuss the phenomenon of Bose-Einstein condensation for a boson gas in three dimensions. In particular, carefully explain why the chemical potential becomes very close to the energy of the lowest single- particle state at sufficiently low temperature and describe how that changes the usual approach of replacing a discrete sum over energies with a continuum integral. Discuss how the occupation of the lowest single-particle state changes as a...
(a) What is the Dulong-Petit Law? Under what eireumstances is it accurate? 2 (b) Einstein's thermal model begins wi...
(a) What is the Dulong-Petit Law? Under what eireumstances is it accurate? 2 (b) Einstein's thermal model begins with the assumption that the average energy for 1 mole of oscillators is given by the expression, Show that this leads to the following expression for the heat capacity Cmol, E/T hwg where E mol (eesT-1) 141 (c) Using the last expression given in part (b) for Cmol, show for the two extremes of the temperature range, (i) In the low temperature...
Please answer the question in full and show all work. We have seen that the absolute...
Please answer the question in full and show all work.
We have seen that the absolute square of the wave function VI,t) can be interpreted as the probability density for the location of the particle at time t. We have also seen that a particle's quantum state can be represented as a linear combination of eigenstates of a physical observable Q: V) SIT) where Q n ) = qn|n) and represents the probability to find the particle in the eigenstate...
Consider two Einstein solids A and B that can exchange energy (but not oscillators/particles) wit...
Using matlab, evaluate the following system:Consider two Einstein solids \(A\) and \(B\) that can exchange energy (but not oscillators/particles) with one another but the combined composite system is isolated from the surroundings. Suppose systems \(A\) and \(B\) have \(N_{A}\) and \(N_{B}\) oscillators, and \(q_{A}\) and \(q_{B}\) units of energy respectively. The total number of microstates for this macrostate for the macrostate \(N_{A}, N_{B}, q, q_{A}\) is given by$$ \Omega\left(N_{A}, N_{B}, q, q_{A}\right)=\Omega\left(N_{A}, q_{A}\right) \Omega\left(N_{B}, q_{B}\right) $$where$$ \Omega\left(N_{i}, q_{i}\right)=\frac{\left(q_{i}+N_{i}-1\right) !}{q_{i} !\left(N_{i}-1\right)...
3 Problem Three [10 points] (The Quantum Oscillator) We have seen in class that the Hamiltonian...
3 Problem Three [10 points] (The Quantum Oscillator) We have seen in class that the Hamiltonian of a particle of a simple Harmonic oscillator potential in one dimension can be expressed in term of the creation and annihilation operators àt and à, respectively, as: or with In >, n = 0,1,..) are the nth eigenstates of the above Hamiltonian. Part A A.1. Show that the energy levels of a simple harmonic oscillator are E,' Aw (nti), n=0, 12, A.2. Calculate...
We have seen in class that at low temperatures, not all dopants are ionized. Consider a...
We have seen in class that at low temperatures, not all dopants are ionized. Consider a Si sample doped with 10^17 cm-3 donors with the location of the donor level ED in the Si bandgap given by EC − ED = 50 meV . Calculate the temperature at which only 50% of the donors are ionized. Assume NC = NV = 2×10^19 cm^-3 and independent of temperature. (b) For the same sample, calculate the high temperature limit of the extrinsic...
(b) For a system of N independent harmonic oscillators at temperature T, all having a common...
(b) For a system of N independent harmonic oscillators at temperature T, all having a common vibrational unit of energy, the partition function is Z = ZN. For large values of N, the system's internal energy is given by U = Ne %3D eBe For large N, calculate the system's heat capacity C. 3. This problem involves a collection of N independent harmonic oscillators, all having a common angular frequency w (such as in an Einstein solid or in the...
Calcium carbonate can decompose into calcium oxide and carbon dioxide gas with the (c) reaction: CaCO3 СаО + СО2 The change in enthalpy for this reaction is AH = +178 kJ/mol and the change in entropy is AS = +0.16 kJ/(K. mol) (i) Describe the concept of the Gibbs energy and how it is related to the enthalpy and entropy (ii Hence calculate the minimum temperature for the decomposition of calcium carbonate to proceed spontaneously An Einstein solid consists of...
[20 pts] It can be shown that the multiplicity of the macrostate with r quanta of energy in an Einstein solid with n oscillators is (a) For an Einstein solid with each of the following values of n and r, list all of the possible microstates, count them, and verify formula E. 1 above. (a-1) n 3,r 5 (a-2) n 3,r-7 (a-3) n 4, r 4 (a-4) n 1,r anything (a-5) n anything,1 (b) Compute the multiplicity of an Einstein...
T2B.2 Calculate the multiplicity of an Einstein solid with N = 1 and U = 5€ by directly listing and counting the microstates. Check your work by using equation T2.7. macrostate of an Einstein solid. R2(0,N) - Vece +3N-1)! (9)! (3N-1)! Write down all of the microstates for a hypothetian HALI 3 Is the
Question 4: (i) Write down the form of the Bose-Einstein distribution and discuss the phenomenon of Bose-Einstein condensation for a boson gas in three dimensions. In particular, carefully explain why the chemical potential becomes very close to the energy of the lowest single- particle state at sufficiently low temperature and describe how that changes the usual approach of replacing a discrete sum over energies with a continuum integral. Discuss how the occupation of the lowest single-particle state changes as a...
(a) What is the Dulong-Petit Law? Under what eireumstances is it accurate? 2 (b) Einstein's thermal model begins with the assumption that the average energy for 1 mole of oscillators is given by the expression, Show that this leads to the following expression for the heat capacity Cmol, E/T hwg where E mol (eesT-1) 141 (c) Using the last expression given in part (b) for Cmol, show for the two extremes of the temperature range, (i) In the low temperature...
Please answer the question in full and show all work.
We have seen that the absolute square of the wave function VI,t) can be interpreted as the probability density for the location of the particle at time t. We have also seen that a particle's quantum state can be represented as a linear combination of eigenstates of a physical observable Q: V) SIT) where Q n ) = qn|n) and represents the probability to find the particle in the eigenstate...
Using matlab, evaluate the following system:Consider two Einstein solids \(A\) and \(B\) that can exchange energy (but not oscillators/particles) with one another but the combined composite system is isolated from the surroundings. Suppose systems \(A\) and \(B\) have \(N_{A}\) and \(N_{B}\) oscillators, and \(q_{A}\) and \(q_{B}\) units of energy respectively. The total number of microstates for this macrostate for the macrostate \(N_{A}, N_{B}, q, q_{A}\) is given by$$ \Omega\left(N_{A}, N_{B}, q, q_{A}\right)=\Omega\left(N_{A}, q_{A}\right) \Omega\left(N_{B}, q_{B}\right) $$where$$ \Omega\left(N_{i}, q_{i}\right)=\frac{\left(q_{i}+N_{i}-1\right) !}{q_{i} !\left(N_{i}-1\right)...
3 Problem Three [10 points] (The Quantum Oscillator) We have seen in class that the Hamiltonian of a particle of a simple Harmonic oscillator potential in one dimension can be expressed in term of the creation and annihilation operators àt and à, respectively, as: or with In >, n = 0,1,..) are the nth eigenstates of the above Hamiltonian. Part A A.1. Show that the energy levels of a simple harmonic oscillator are E,' Aw (nti), n=0, 12, A.2. Calculate...
(b) For a system of N independent harmonic oscillators at temperature T, all having a common vibrational unit of energy, the partition function is Z = ZN. For large values of N, the system's internal energy is given by U = Ne %3D eBe For large N, calculate the system's heat capacity C. 3. This problem involves a collection of N independent harmonic oscillators, all having a common angular frequency w (such as in an Einstein solid or in the...
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