Question

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" $\mu =-kT\ln \left(\frac{N+q}{N}\right)$

(b) Discuss this result in the limits $N\gg q$and $N\ll q$, 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

(a) It is proved that the chemical potential is $\mu =-kT\ln \left(\frac{N+q}{N}\right)$

(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

$\begin{array}{r}\mathrm{\Omega }=\sqrt{\frac{N}{2\pi q(N+q)}}{\left(\frac{q+N}{q}\right)}^q{\left(\frac{q+N}{N}\right)}^N\\ \mathrm{\Omega }\approx {\left(\frac{q+N}{q}\right)}^q{\left(\frac{q+N}{N}\right)}^N\end{array}$

The entropy is

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

The chemical potential is

$\begin{array}{r}\mu =-T\left(\frac{\mathrm{\partial }S}{\mathrm{\partial }N}\right)U.V\\ \mu =-Tk\frac{\mathrm{\partial }}{\mathrm{\partial }N}\left[\right(q+N)\ln (q+N)-q\ln q-N\ln N]\\ \mu =-kT\ln \left(\frac{q+N}{N}\right)\end{array}$

Part (a) Step 3 : Conclusion

The chemical potential is,

$\mu =-kT\ln \left(\frac{q+N}{N}\right)$

Part (b) Step 1 : Given Information

$$\begin{gathered} N>>q \\ N<<q \\ U=0 \end{gathered}$$

Part (b) Step 2 : Calculation

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

$\mu \approx -kT\ln \left(1+\frac{q}{N}\right)$

At $N\ll q$, the chemical potential is,

$\begin{array}{r}\mu \approx -kT\ln \left(\frac{q}{N}\right)\\ \mu \to -\mathrm{\infty }\end{array}$

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.