Chapter 3: Q. 3.36 (page 119)

Consider an Einstein solid for which both N and q are much greater than $1$ . Think of each oscillator as a separate "particle."
(a) Show that the chemical potential is
role="math" localid="1646995468663" $μ = - k T \ln (\frac{N + q}{N})$
(b) Discuss this result in the limits $N ≫ q$ and $N ≪ q$ , concentrating on the question of how much $S$ increases when another particle carrying no energy is added to the system. Does the formula make intuitive sense?

Short Answer

Expert verified

(a) It is proved that the chemical potential is $μ = - k T \ln (\frac{N + q}{N})$

(b) When $N > > q$ there are more oscillation than energy quanta, so the multiplicity is about the same.

When $N < < q$ there are more quanta than oscillator so there is more possible microstate. So, the multiplicity and the entropy increases.

Step by step solution

Part (a) Step 1 : Given Information

Number of oscillations, $N > > 1$

Number of units of energy, $q > > 1$

Part (a) Step 2 : Calculation

The multiplicity is

$\begin{aligned} Ω = \sqrt{\frac{N}{2 π q (N + q)}} {(\frac{q + N}{q})}^{q} {(\frac{q + N}{N})}^{N} \\ Ω \approx {(\frac{q + N}{q})}^{q} {(\frac{q + N}{N})}^{N} \end{aligned}$

The entropy is

$S = k [(q + N) \ln (q + N) - q \ln q - N \ln N]$

The chemical potential is

$\begin{aligned} μ = - T (\frac{\partial S}{\partial N}) U . V \\ μ = - T k \frac{\partial}{\partial N} [(q + N) \ln (q + N) - q \ln q - N \ln N] \\ μ = - k T \ln (\frac{q + N}{N}) \end{aligned}$

Part (a) Step 3 : Conclusion

The chemical potential is,

$μ = - k T \ln (\frac{q + N}{N})$

Part (b) Step 1 : Given Information

$N > > q N < < q U = 0$

Part (b) Step 2 : Calculation

At $N > > q$ , the chemical potential is,

$μ \approx - k T \ln (1 + \frac{q}{N})$

At $N ≪ q$ , the chemical potential is,

$\begin{aligned} μ \approx - k T \ln (\frac{q}{N}) \\ μ \to - \infty \end{aligned}$

Part (b) Step 3 : Conclusion

When $N > > q$ there are more oscillation than energy quanta, so the multiplicity is about the same. When $N < < q$ there are more quanta than oscillator so there is more possible microstate. So, the multiplicity and the entropy increases.