Chapter 6: Q 6.2 (page 224)

Prove that the probability of finding an atom in any particular energy level is $P (E) = (1 / Z) e^{- F / k T}$ , where $F = E - T S$ and the "'entropy" of a level is k times the logarithm of the number of degenerate states for that level.

Short Answer

Expert verified

Hence proved that the probability of finding an atom in any particular energy level is: $P (E) = (1 / Z) e^{- F / k T}$

Step by step solution

Given information

The probability of finding an atom in any particular energy level is

$P (E) = (1 / Z) e^{- F / k T}$ where $F = E - T S$

Explanation

Assume we have an n-degenerate level; the chance of a system being in that level is just n multiplied by the probability of being in any of the states, as follows:

$P (E) = n P (s)$

From equation 6.8, for any state we have:

$P (s) = \frac{1}{Z} e^{- E (s) / k T}$

Substitute into the above equation

$P (E) = \frac{1}{Z} n e^{- E (s) / k T} (1)$

The entropy of the system will be:

$S = k \ln (n) \ln (n) = \frac{S}{k} n = e^{S / k}$

Calculations

Substitute the value of n in equation (1)

$P (E) = \frac{1}{Z} e^{S / k} e^{- E (s) / k T} P (E) = \frac{1}{Z} e^{S / k - E (s) / k T} P (E) = \frac{1}{Z} e^{T S / k T - E (s) / k T} P (E) = \frac{1}{Z} e^{(T S - E (s)) / k T}$

Using $F = E (s) - T S$ where F is the Helmholtz free energy:

$P (E) = \frac{1}{Z} e^{- F / k T}$

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Prove that the probability of finding an atom in any particular energy level is $P (E) = (1 / Z) e^{- F / k T}$ , where $F = E - T S$ and the "'entropy" of a level is k times the logarithm of the number of degenerate states for that level.

Short Answer

Step by step solution

Given information

Explanation

Calculations

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Prove that the probability of finding an atom in any particular energy level is P(E)=(1/Z)e-F/kT, whereF=E-TS and the "'entropy" of a level is k times the logarithm of the number of degenerate states for that level.

Short Answer

Step by step solution

Given information

Explanation

Calculations

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Rotational Dynamics

Wave Optics

Electricity

Measurements

Work Energy and Power

Fields in Physics

Study anywhere. Anytime. Across all devices.

Prove that the probability of finding an atom in any particular energy level is $P (E) = (1 / Z) e^{- F / k T}$ , where $F = E - T S$ and the "'entropy" of a level is k times the logarithm of the number of degenerate states for that level.