**Einstein and Molar Heat Capacity of crystals (breakdown of classical mechanics)**

In classical mechanics, the heat capacity is the ability of a crystalline solid to absorb and retain heat. In 1819 Pierre Dulong and Alexis Petit decided to investigate this quality in a variety of solids, and they found that for all solids investigated the heat capacity was relatively the same at room temperature (298K). They explained their findings through the idea that every atom in the lattice has six degrees of freedom, or directions in which they can vibrate when they are excited. Mathematically they used the following equation to model the specific heat:

C_{v}=3Nk

Where N=the number of atoms in the solid

And k=an arbitrary constant

The Petit and Dulong model and value for the molar heat capacity of solids did a great job of modeling the molar heat capacity of solids, except for when temperatures got very low (approaching absolute zero) and the molar heat capacity dropped off sharply. Dulong and Petit were unable to explain this change since, according to their model, specific heat was based on degrees of freedom and should be independent of temperature.

Several years later Einstein sought to explain the strange behavior of solids in terms of heat capacity at low temperatures. He attempted to do this by adapting Planck’s ideas about black body radiation as oscillating particles. Einstein imagined a solid as a group of atoms or molecules all oscillating at the same frequency v_{e }This frequency came to be known as the Einstein frequency. Einstein postulated that the energy of these oscillators is equal to:

E_{v}=hv_{e}(v+1/2) (v=1,2,3…)

As you can see, much like in Planck’s black bodies, the energy level of the oscillators can only increase in discrete quanta “packets.” Einstein then factored this energy level into a final equation for the specific heat of solids:

C_{v}=3Nk (x^{2}e^{x})/(e^{x}-1)^{2 }

Where x=hv_{e}/T

As you can see when temperatures are close to zero x becomes very large, and the quantized energy state plays a large role in the specific heat, however when temperatures rise x becomes small and the specific heat begins to resemble the Dulong and Petit model.

**Shortcomings of the Quantum Model**

Due to the various scientific discoveries that we have discussed, scientists are now able to quite accurately model the behavior of particles down to the quantum level. To make things easier to think about, most of these models and equations were discovered by using the most basic variables available, namely the hydrogen atom. Since it only has one electron, scientists found that observing the hydrogen atom was the easiest way to learn more about quantum behavior. While this proved to be true, and we now have many equations and models to explain quantum behavior, scientists quickly found that when using more complicated atoms the equations have the possibility to become too complicated to solve. For example, the Schrodinger equation is a great way to model the position or momentum of electrons, however due to its complex Hamiltonian operator, scientists so far have been unable to solve this equation for anything other than hydrogen, and “hydrogen like atoms” (ions with only one electron He^{+}, Li^{2+}…etc) on one hand this is disappointing because we are still unable to describe a large amount of quantum behavior but on the other it is exciting because there is still so much left to discover.

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