If we expand the energy around a variable $x_i$:

$dE = -\displaystyle\frac{\partial E}{\partial x_i}dx_i$

we recognize that the derivative of energy with respect to this variable acts as a force that tends to resist changes in the variable. For this reason, the derivative of force with respect to the variable $x_i$, with list

is called the generalized force. The generalized force is an intensive variable (it does not depend on the system's size), while the associated variable is an extensive variable (it depends on the system's size).

An example of an extensive variable is volume. When we consider a larger system, its volume increases. However, pressure is intensive, meaning it does not increase when considering a larger system.

As the generalized force e with list=3445s

can be rewritten as list=3442

$X_i =\displaystyle\frac{\partial E}{\partial x_i}=\displaystyle\frac{\partial E}{\partial S}\displaystyle\frac{\partial S}{\partial x_i}=T\displaystyle\frac{\partial}{\partial x_i} (k\ln\Omega)$

which results in list=3437

with list

An article that summarizes most of the thermodynamic relationships very well is Use of Legendre Transforms in Chemical Thermodynamics, Robert A. Alberty, Pure Appl. Chem., Vol. 73, No. 8, pp. 13491380, 2001 containing the following table of pairs of extensive and intensive variables:

image

An example of an extensive variable and generalized force is the volume $V$ with pressure $p$. In this case, the relationship for the generalized force is given with list as follows:

Si empleamos el numero de estados para el caso de un gas ideal tendremos que el numero de estados es con list=4805

$\displaystyle\frac{\partial \ln\Omega}{\partial V}=\displaystyle\frac{N}{V}$

por lo que se obtiene con list: