Chapter 9: Q32E (page 404)

Using the relationship between temperature and $M$ and $N$ given in (9-16) and that between $E$ and $n$ in (9-6), obtain equation (9-17) from (9- 12). The first sum given In Exercise 30 will be useful.

Short Answer

Expert verified

The expression for $P (E_{n})$ is $\frac{N}{N + M} e^{- \ln (1 + \frac{N}{M})}$

Step by step solution

Boltzmann Distribution.

The Boltzmann distribution for a system of particles in terms of n, M, and N is given by:

$P (E_{n}) = \frac{N}{M + N} e^{- n l n (1 + \frac{N}{M})}$

Boltzmann Probability in terms of $E_{n}$ and $k_{B} T$ :

$P (E_{n}) = \frac{e^{\frac{- E_{n}}{k_{B} T}}}{_{n = 0}^{\infty} e^{\frac{- E_{n}}{k_{B} T}}}$

System of N Harmonic Oscillators.

$\begin{matrix} E_{n} = n h ω_{0} \\ P (E_{n}) = \frac{e^{\frac{- n h ω_{0}}{k_{B} T}}}{\sum_{n = 0}^{\infty} e^{\frac{- n h ω_{0}}{k_{B} T}}} \\ = \frac{e^{\frac{- n h ω_{0}}{k_{B} T}}}{\sum_{n = 0}^{\infty} {[e^{\frac{- n h ω_{0}}{k_{B} T}}]}^{n}} \end{matrix}$

Properties.

$\sum_{n = 0}^{\infty} x^{n} = \frac{1}{1 - x}$

Denominator of P:

$\begin{matrix} P (E_{n}) = e^{\frac{- n h ω_{0}}{k_{B} T}} \frac{1}{1 - e^{\frac{- n h ω_{0}}{k_{B} T}}} \\ = (1 - e^{^{\frac{- n h ω_{0}}{k_{B} T}}}) e^{^{\frac{- n h ω_{0}}{k_{B} T}}} \\ k_{B} T = \frac{h ω_{0}}{\ln (1 + \frac{N}{M})} \\ \frac{h ω_{0}}{k_{B} T} = \ln (1 + \frac{N}{M}) \end{matrix}$

On further calculation,

$\begin{matrix} P (E_{n}) = (1 - e^{- \ln (1 + \frac{N}{M})}) e^{^{- \ln (1 + \frac{N}{M})}} \\ = 1 - \frac{1}{e^{^{- \ln (1 + \frac{N}{M})}}} e^{^{- \ln (1 + \frac{N}{M})}} \\ = 1 - \frac{1}{1 + \frac{N}{M}} e^{^{- \ln (1 + \frac{N}{M})}} \\ = \frac{1 + \frac{N}{M} - 1}{1 + \frac{N}{M}} e^{- \ln (1 + \frac{N}{M})} \\ = \frac{\frac{N}{M}}{\frac{(M + N)}{M}} e^{- \ln (1 + \frac{N}{M})} \\ = \frac{N}{N + M} e^{- \ln (1 + \frac{N}{M})} \end{matrix}$

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Using the relationship between temperature and $M$ and $N$ given in (9-16) and that between $E$ and $n$ in (9-6), obtain equation (9-17) from (9- 12). The first sum given In Exercise 30 will be useful.

Short Answer

Step by step solution

Boltzmann Distribution.

System of N Harmonic Oscillators.

Properties.

One App. One Place for Learning.

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Using the relationship between temperature and MandN given in (9-16) and that betweenE andn in (9-6), obtain equation (9-17) from (9- 12). The first sum given In Exercise 30 will be useful.

Short Answer

Step by step solution

Boltzmann Distribution.

System of N Harmonic Oscillators.

Properties.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Nuclear Physics

Electricity

Classical Mechanics

Thermodynamics

Rotational Dynamics

Further Mechanics and Thermal Physics

Study anywhere. Anytime. Across all devices.

Using the relationship between temperature and $M$ and $N$ given in (9-16) and that between $E$ and $n$ in (9-6), obtain equation (9-17) from (9- 12). The first sum given In Exercise 30 will be useful.