1. Consider the system of $N$ classical harmonic oscillators. (a) Using microcanonical ensemble, compute the number...
1. Consider the system of $N$ classical harmonic oscillators.
(a) Using microcanonical ensemble, compute the number of accessible state $Omega(E,N)$, where $E$ is the total energy of the system.
(b) Find the expression for the heat capacity $C(T)$ as a function of the temperature, and draw its graph.
(c) Using canonical ensemble, compute the partition function $Z(T,N)$ of the system.
(d) Repeat b) in canonical ensemble.
2. Repeat Problem 1 for the system of $N$ (distinguishable) quantum harmonic oscillators.
Homework Answers
Add Answer to:
1. Consider the system of $N$ classical harmonic
oscillators.
(a) Using microcanonical ensemble, compute the number...
1: Consider a system of N classical distinguishable one-dimensional harmonic oscillators with fre...
1: Consider a system of N classical distinguishable one-dimensional harmonic oscillators with frequency w in the microcanonical ensemble. Determine the phase space volume and the corresponding entropy. Please be sure to include the quantum correction h3N. Please note that the Hamiltonian for the system isN mwqi You may also want to note that the volume of a d-dimensional which can be simplified by using r sphere of radius R is given by
1: Consider a system of N classical distinguishable...
1. (10 pts) Consider a system of N classical, independent harmonic oscil- lators. In the microcanonical...
1. (10 pts) Consider a system of N classical, independent harmonic oscil- lators. In the microcanonical ensemble, calculate Ω(E) and Ω(B) exactly. From them, calculate the entropy S(E, N) and temperature T in the large N limit. 2. (10 points) Consider the same system as in problem 1. Calculate the average energy and entropy starting from the canonical ensemble.
(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...
In the canonical ensemble, a computation of the N-particle partition function in a particular ex- ample...
In the canonical ensemble, a computation of the N-particle partition function in a particular ex- ample reveals that ZN = temperature T. What is Cv, the specific heat at constant volume, for this system? aVNT3N, where a is a constant independent of the volume V and
Classically, the heat capacity of a chain of N 1-dimensional harmonic oscillators equals N k, i.e.,...
Classically, the heat capacity of a chain of N 1-dimensional harmonic oscillators equals N k, i.e., it is temperature independent. Explain qualitatively why the heat capacity should decrease as T goes to 0 K.
Pb2. Consider the case of a canonical ensemble of N gas particles confined to a t...
Pb2. Consider the case of a canonical ensemble of N gas particles confined to a t rectangular parallelepiped of lengths: a, b, and c. The energy, which is the translational kinetic energy, is given by: o a where h is the Planck's constant, m the mass of the particle, and nx, ny ,nz are integer numbers running from 1 to +oo, (a) Calculate the canonical partition function, qi, for one particle by considering an integral approach for the calculation of...
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 1 5 points] In classical statistical mechanics, the canonical partition function for a single harmonic...
PROBLEM 1 5 points] In classical statistical mechanics, the canonical partition function for a single harmonic oscillator is of the form d dp e Δ ΔΊΔ ) is the regulating spatial and momentum resolution cutoffs, which are often Chosen to be at the scale of the atoms (and n) and are important for making entropy dimensionless but they drop out in parts (b) and (c). Moreover, Z factorizes as Z ZzZp with Z. 3 Calculate the partition function and the...
Consider a classical particle confined on a segment with length 2L. If a harmonic potential is...
Consider a classical particle confined on a segment with length 2L. If a harmonic potential is introduced on this segment, then the Hamiltinian becomes and application of the equipartition theorem predicts the average values of the kinetic and potential energies, e.g., <V>-kT/2. On the hand, making L sufficiently small, one can ensure that V(x)=ma,2x2/2ckT/2 for all x between-L and L. We conclude that the average value is larger than any allowed value of V! Please explain this paradox (assume that...
question no 4.22, statistical physics by Reif Volume 5 4.92 Mean energy of a harmonic oscillator A harmonic oscillator...
question no 4.22, statistical physics by Reif Volume 5
4.92 Mean energy of a harmonic oscillator A harmonic oscillator has a mass and spring constant which are such that its classical angular frequency of oscllation is equal to w. In a quantum- mechanical description, such an oscillator is characterized by a set of discrete states having energies En given by The quantum number n which labels these states can here assume all the integral values A particular instance of a...
1: Consider a system of N classical distinguishable one-dimensional harmonic oscillators with frequency w in the microcanonical ensemble. Determine the phase space volume and the corresponding entropy. Please be sure to include the quantum correction h3N. Please note that the Hamiltonian for the system isN mwqi You may also want to note that the volume of a d-dimensional which can be simplified by using r sphere of radius R is given by
1: Consider a system of N classical distinguishable...
1. (10 pts) Consider a system of N classical, independent harmonic oscil- lators. In the microcanonical ensemble, calculate Ω(E) and Ω(B) exactly. From them, calculate the entropy S(E, N) and temperature T in the large N limit. 2. (10 points) Consider the same system as in problem 1. Calculate the average energy and entropy starting from the canonical ensemble.
(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...
In the canonical ensemble, a computation of the N-particle partition function in a particular ex- ample reveals that ZN = temperature T. What is Cv, the specific heat at constant volume, for this system? aVNT3N, where a is a constant independent of the volume V and
Pb2. Consider the case of a canonical ensemble of N gas particles confined to a t rectangular parallelepiped of lengths: a, b, and c. The energy, which is the translational kinetic energy, is given by: o a where h is the Planck's constant, m the mass of the particle, and nx, ny ,nz are integer numbers running from 1 to +oo, (a) Calculate the canonical partition function, qi, for one particle by considering an integral approach for the calculation of...
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 1 5 points] In classical statistical mechanics, the canonical partition function for a single harmonic oscillator is of the form d dp e Δ ΔΊΔ ) is the regulating spatial and momentum resolution cutoffs, which are often Chosen to be at the scale of the atoms (and n) and are important for making entropy dimensionless but they drop out in parts (b) and (c). Moreover, Z factorizes as Z ZzZp with Z. 3 Calculate the partition function and the...
Consider a classical particle confined on a segment with length 2L. If a harmonic potential is introduced on this segment, then the Hamiltinian becomes and application of the equipartition theorem predicts the average values of the kinetic and potential energies, e.g., <V>-kT/2. On the hand, making L sufficiently small, one can ensure that V(x)=ma,2x2/2ckT/2 for all x between-L and L. We conclude that the average value is larger than any allowed value of V! Please explain this paradox (assume that...
question no 4.22, statistical physics by Reif Volume 5
4.92 Mean energy of a harmonic oscillator A harmonic oscillator has a mass and spring constant which are such that its classical angular frequency of oscllation is equal to w. In a quantum- mechanical description, such an oscillator is characterized by a set of discrete states having energies En given by The quantum number n which labels these states can here assume all the integral values A particular instance of a...
Most questions answered within 3 hours.
-
what is the product when Piperidine is reacted with NaNO2,
HCl?
asked 1 hour ago -
a restaurant offers a 15$ dinner special that lets you choose
from 6 appetizers, 12 entrees,...
asked 1 hour ago -
What is the advantage of using steam distillation to isolate
essential oils, as opposed to solvent...
asked 2 hours ago -
A person slams their car door on the other side of a parking
lot, 150 m...
asked 1 hour ago -
Give the following balanced chemical equation of the
following:
1. Assay of Tartaric Acid
2. Standardization...
asked 1 hour ago -
You and a group of friends are playing mini golf. You have to
hit a ball...
asked 2 hours ago -
An employee works 51 hours (51 - 40 were overtime hours) during
a workweek in December...
asked 2 hours ago -
At the present time, generating energy from the tides does not
make a significant contribution to...
asked 2 hours ago -
For a regular language below, write an equivalent regular
expression and draw an FA that accepts...
asked 2 hours ago -
4. My VERY FANCY LUNCHBOX always maintains itself at the SAME
TEMPERATURE and the SAME VOLUME....
asked 2 hours ago -
1. Rank the reactivity of acetophenone, 4-bromoacetopheone, and
4-chloroacetopheone when mixed with benzaldehyde. Which will be...
asked 2 hours ago -
In Drosophila, glued eyes is due to a dominant
allele Ge that is lethal when homozygous....
asked 2 hours ago