Answered: Problem 1: Consider a classical ideal… | bartleby

Related questions

Question

Problem 1:
Consider a classical ideal gas in three dimensions, with N indistinguishable
atoms confined in a box of volume N³. Assume the atoms have zero spin
and neglect any internal degrees of freedom. Starting from the energy levels
of a single atom in a box, find:
(a) The Helmholtz free energy F'
Hint: ſ.
-ax²
d.x
e
Va
(b) The entropy o
(c) The pressure p

Expert Solution

Trending now

This is a popular solution!

Step by step

Solved in 3 steps

SEE SOLUTION Check out a sample Q&A here

Blurred answer

Similar questions

When developing the partition functions for each degree of freedom in a diatomic ideal gas, we made some approximations. Match the degree of freedom with the approximation(s) we made. Classical (energy levels are continuous) and the addition of one extra state. Classical (energy levels are continuous) no approximations excited states do not contribute significantly [Choose ] [Choose ] [Choose ] [Choose ] ♦
Consider a classical ideal gas of N diatomic heterogeneous molecules at temperature T. The charac- teristic rotational energy parameter is € = 1 and the natural frequency of vibrations is wo. Consider the temperature region where T≫er/kB, but T is of the order of ħwo/kB. Ignore contributions from all other internal modes. Calculate the canonical partition function, the average energy, and the heat capacity at constant volume, Cv.
Show that the one-particle partition function Z₁ for a 2D ideal gas confined to area A is: A 2²/1 Z₁ = S
Q2/ For the harmonic oscillator in classical limit, if the partition function is given by Z(B) = 1/(ßħw)N, find the entropy and internal energy. Q3/ Prove that F = -KTlnz .
Calculate the partition function of a two-level system at 25 °C with an energy gap of 10-2¹ J, assuming: a) Both states are non-degenerate. b) The ground state is non-degenerate, and the excited state is 3-fold degenerate.
Plot the Fermi-Dirac probability of occupation function fFD(E) for T = 0, 10, 100, 200, 300 and 400K.
The exact differential for the Gibbs energy is given by dG = -SdT + VdP. The form of this differential implies which of the following relationships? O (7),- (), ƏG Әт ƏG др P T (37), - - (-), P T as ( x) - (*), = др T (327), = -(0)₁ == P T
A diatomic gas molecule can be in one of two vibrational energy levels, separated by 0.1 eV. Give the probabilities to be in either state and use these to calculate their relative populations at room temperature, T≈ 300 K. [You may use that kB ≈ 8.6 × 10−5 K eV−1]
Consider a molecular two-level system with a twofold degenerate ground state (la- beled "0") and a threefold degenerate level (labeled “1") with an energy (expressed in wavenumbers) 150.0 cm-1 above the ground state. (a) What is the proportion N1/N of molecules in the higher level at T = 400 K? (b) What is the molecular average energy (relative to the ground state and expressed in wavenumbers) when the temperature is T = 400 K? (c) What is the proportion No/N of molecules in the ground state when T → 0K? (d) What is the proportion N1/N of molecules in the higher level when T → x?