Propiedades térmicas#
Dulong-Petit#
(422)#\[C_V\approx 3N_A K =3R =24.94 J/mol K\]
Modelo Einstein#
(423)#\[U=\sum_{a\vec{k}} n_{a\vec{k}} \hbar\omega_{a\vec{k}}\]
(424)#\[U=\sum_{a\vec{k}} n_{a\vec{k}} \hbar\omega_{a\vec{k}}\approx 3N n_{a\vec{k}} \hbar\omega =\frac{3N \hbar\omega}{e^{\frac{\hbar\omega}{KT}}-1}\]
(425)#\[\begin{split}C_V=\left.\frac{dU}{dT}\right\vert_V=3NK\left(\frac{\hbar\omega}{KT}\right)^2\frac{e^{\frac{\hbar\omega}{KT}}}{\left[e^{\frac{\hbar\omega}{KT}}-1 \right]^2} \rightarrow
\begin{cases}
KT >> \hbar \omega ~~~~~ c_v \approx 3N_A K ~~~~~ \text{Dulong-Petit}\\
KT << \hbar \omega ~~~~~ c_v \approx \frac{3N_AK (\hbar\omega)^2}{K^2} \frac{1}{T^2} \neq \frac{1}{T^3}
\end{cases}\end{split}\]
Definición#
The divergence is an operator that is applied on a vector field.
(426)#\[\vec{\nabla}\cdot \vec{f}=\frac{\partial f_x}{\partial x}+\frac{\partial f_y}{\partial y}+\frac{\partial f_z}{\partial z}\]
As can be seen, the result of the applying the divergence is a scalar field.
Interpretation#
The divergence measures the difference between the number of field lines that enter and leave from/to a particular point.
Ejemplo
Fig. 211 The divergence measures the number of lines entering/exiting a point.#
A positive divergence is associated with a majority of lines coming out (leaving) from a point. This point is a source of field lines. On the other hand, a point with a negative divergence is seen to come from a majority of lines entering into a point. This point is called a sink of field line.