Question

7. The statistical thermodynamics of harmonic oscillations. Write the internal energy, entropy, enthalpy, free energy, and pressure for a system of N independent distinguishable harmonic oscillators.

Answer 1

For harmonic oscillators,

hes q=el--M8)

=Planck's constant

$\nu$=vibrational frequency

For N distinguishable paricles,

wbro
=$\left ( e^{\frac{-h\nu \beta }{\left ( 1-e^{-h\nu \beta } \right )}} \right )^{n}$

Internal energy:

AU 180 hve-hu N = qaB = 1- e-hus

Entropy:

$\frac{S}{kN}= ln q+\frac{\Delta U}{NkT}$

hu) e-huß =-In (1 - e-hus) + KT 1-e-hus

T= Temperature of oscillator

Free energy:

F= -kTinQ

F T=-Inq = +In (1 - e-hus)

Pressure and Enthalpy:

It's unclear to discuss on pressure and enthalpy.

To truly depend on pressurea harmonic oscillator would have to have volume. In that sense a harmonic oscillator is independent of pressure-volume and pressure simply don't appear as variables. (we can control oscillator size by changing its frequen