Somehow you have a two-dimensional solid, a sheet of atoms in a square lattice, each atom linked to its four closest neighbors by four springs oriented along the two perpendicular axes. (a) What would you expect the molar heat capacity to be at very low temperatures and at very high temperatures? (b) What quantity would determine, roughly, the line between low and high?

Molarity of solution is,

….. (1)

Here, is the number of the degrees of freedom (on which its energy depends quadratically),$\mathbf{R}$ is the gas constant, and$\mathbf{\text{T}}$ is the temperature.

${\mathbf{U}}_{\mathbf{mol}}\mathbf{=}\mathbf{n}\mathbf{2}\mathbf{RT}$

The equation for the molar heat capacity${\mathbf{\text{C}}}_{\mathbf{v}}$ is also needed:

${\mathbf{\text{C}}}_{\mathbf{v}}\mathbf{=}\raisebox{1ex}{$\mathbf{\partial }\mathbf{\text{U}}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{\partial }\mathbf{\text{T}}$}\right.$

This represents that it's equal to the partial derivative of the energy$\mathbf{\text{U}}$ with respect to temperature $\mathbf{T}$.

For high temperatures, apply equation (2) to equation (1) to find the molar heat capacity:

$\begin{array}{c}{\text{C}}_{\mathrm{v}}=\raisebox{1ex}{$\partial \mathrm{U}$}\!\left/ \!\raisebox{-1ex}{$\partial \mathrm{T}$}\right.\\ =\frac{\partial }{\partial \mathrm{T}}\left(\mathrm{n}2\mathrm{RT}\right)\\ =\mathrm{n}2\mathrm{R}\end{array}$

To as certain that value, identify the number of degrees of freedom for the 2-D material.

Since it has velocity in two dimensions, it gives two degrees of freedom (since the kinetic energy depends on velocity squared).

However, there is also energy from the springs in each dimension (since the potential energy goes as distance squared), which gives another two degrees of freedom. So there's a total of 4 degrees of freedom, which is inserted in for in the previous equation:

$\begin{array}{c}{C}_v=n2R\\ =\left(4\right)2R\\ =8R\end{array}$

So at high temperatures, it would be expected that the molar heat capacity for the 2-D solid would be:

${\mathrm{C}}_{\mathrm{v}}=8\text{R}$

For low temperatures, it would be expected that the molar heat capacity would be approximately $0$ because the degrees of freedom that were available at high temperatures are effectively unavailable to store energy; the atoms need some freedom of movement to be able to use the previously mentioned degrees of freedom, the would be $0$, and thus the ${\mathrm{C}}_{\mathrm{v}}$ would be approximately $0$.

The quantity that is used to determine the difference between a "high" or "low" temperature is the Debye temperature of the material.

If the temperature of the material is approximately the same as its Debye temperature, the temperature is considered "high".

If the temperature is less than roughly of the Debye temperature, it's considered "cold". The temperature between those two cut-offs gives a steadily increase in heat capacity.