Question

Calculate the multiplicity of an Einstein solid with $30$oscillators and $30$units of energy. (Do not attempt to list all the microstates.)

Short answer

An Einstein solid with $30$oscillators and $30$energy units has a multiplicity of$\mathrm{\Omega }(30,30)=591322907824307$

Step-by-step solution

Step1:The number of oscillators contained in an Einstein solid.

Using the general Einstein solid multiplicity formula with$N=3$and $q=4$:

$\begin{array}{c}\mathrm{\Omega }(N,q)=\left(\begin{array}{c}q+N-1\\ q\end{array}\right)=\frac{(q+N-1)!}{q!(N-1)!}\\ \mathrm{\Omega }(3,4)=\left(\begin{array}{c}4+3-1\\ 4\end{array}\right)=\frac{(4+3-1)!}{4!(3-1)!}=15\end{array}$

Step2:The total number of microstates

So, if we have$30$oscillators and energy units, the number of$30$microstates is:

$$\begin{gathered} \mathrm{\Omega }(30,30)=\left(\begin{array}{c}30+30-1\\ 30\end{array}\right) \\ =\frac{(30+30-1)!}{30!(30-1)!} \\ =591322907824307 \end{gathered}$$

$\mathrm{\Omega }(30,30)=591322907824307$