### Exercise 3: Total heat capacity of a diatomic material
Naturally occurring lithium has[two stable isotopes](https://en.wikipedia.org/wiki/Isotopes_of_lithium): $^6$Li (7.5%) and $^7$Li (92.5%). Let us extend the Einstein model to take into account the different masses of different isotopes.
We consider a crystal of lithium, which consists of the[two stable isotopes](https://en.wikipedia.org/wiki/Isotopes_of_lithium) $^6$Li (7.5%) and $^7$Li (92.5%) in their natural abundance. Let us extend the Einstein model to take into account the different masses of these different isotopes.
1. Assume that the strength of the returning force $k$ experienced by each atom is the same. What is the difference in the oscillation frequencies of different isotopes of lithium in the lithium crystal?
2. Write down the total energy of lithium assuming that all $^6$Li atoms are in $n=2$ vibrational state, and all $^7$Li atoms are in $n=4$ vibrational state.
3. Write down the total energy of lithium at a temperature $T$ by modifying the Einstein model.
4. Compute the heat capacity of lithium as a function of $T$.
1. Assume that the strength of the returning force $k$ experienced by each atom is the same. What is the difference in the oscillation frequencies of the two different isotopes in the lithium crystal?
2. Write down the total energy stored in the vibrations of the atoms of the lithium crystal, assuming that all $^6$Li atoms are in $n=2$ vibrational state and all $^7$Li atoms are in $n=4$ vibrational state.
3. Write down the total energy stored in the vibrations of the atoms in the lithium crystal at a temperature $T$ by modifying the Einstein model.
4. Compute the heat capacity of the lithium crystal as a function of $T$.
[^1]:Data source: [Wikipedia](https://en.wikipedia.org/wiki/Heat_capacities_of_the_elements_(data_page)), mainly the CRC Handbook of Chemistry and Physics.
