# Chemistry - Is zero-point vibrational energy an intensive or extensive property?

## Solution 1:

Mathematically, an extensive property $f$ is one for which $f(\lambda x,\lambda y,z) = \lambda f(x,y,z)$ for extensive variables $x$, $y$, and intensive variable $z$. For example, the Helmholtz free energy $A(T,\lambda V,\lambda N) = \lambda A(T,V,N)$ is extensive.

Following this definition, the zero-point energy $$E_\text{ZPVE}(\{\omega_c\},\lambda n) = \lambda E_\text{ZPVE}(\{\omega_c\},n),$$ where $n$ is the number of normal modes, is extensive in the thermodynamic limit $N \to \infty$.

The polarizability should be able to be treated similarly.

It bears noting that a lot of macroscopic extensive properties are not extensive in the microscopic regime. For example, if we consider two identical systems, each with an energy $U$, then the combined system has energy $2U$ plus an additional energetic term $U_\text{sys-sys}$, accounting for superficial energetic interactions between the two subsystems. The surface, and hence the superficial energy term, scales as one dimension removed of the volume, and likewise vanishes in the thermodynamic limit.

Furthermore, it's perfectly valid to have properties that are neither extensive nor intensive, like the number of microstates that a system possesses.

## Solution 2:

Is the zpe not a term in U just like other vibrational and rotational energy levels? The word 'internal' is confusing in molecular terms as it refers to a thermodynamic 'system' not molecules (which do not need to exist as far as thermodynamics goes). The internal energy is defined by First Law and so for an ideal/perfect gas of point like atoms it can have no other energy than from translational motion. After it was discovered that molecule have vibrational/rotational energy then it is possible that these energies can be changed by affecting the heat absorbed by and work done on the 'system' and so are part of the system's internal energy.

The point about intensive variables is that their value does not depend on how much 'stuff' there is, e.g. temperature and pressure.