Question

how are extensive and intensive energies related

Answer 1

Intensive energy is derived from extensive energy by dividing by another extensive property like moles or mass, normalizing it to be independent of system size.

Answer 2

Extensive energies scale with the amount of matter (e.g., total internal energy, total enthalpy). Intensive energies do not scale with the amount of matter (e.g., molar internal energy, specific enthalpy). The relationship is that an intensive energy is obtained by dividing an extensive energy by another extensive quantity, such as moles or mass. This effectively normalizes the energy per unit amount, making it independent of the system size.

Mathematically, if $X_{ext}$ is an extensive energy (like Internal Energy, $U$) and $N_{ext}$ is an extensive quantity (like moles, $n$, or mass, $m$), then the corresponding intensive energy $X_{int}$ is:

$X_{int} = \frac{X_{ext}}{N_{ext}}$

For example:

• Molar Internal Energy ($U_m$): An intensive property obtained by dividing the extensive total internal energy ($U$) by the extensive number of moles ($n$).

  $U_m = \frac{U}{n}$

• Specific Enthalpy ($h$): An intensive property obtained by dividing the extensive total enthalpy ($H$) by the extensive mass ($m$).

  $h = \frac{H}{m}$

This conversion is crucial for comparing energy properties of different systems or the same substance under varying quantities.