Normal Mode Calculation of Grüneisen Thermal Expansion in n-Alkanes

Abstract

The normal mode problem is solved for a semirigid chain of beads in a quasiharmonic potential in order to separate the inter‐ and intrachain parts of the specific heat. The effects of lattice vibrations are included in an approximate way. The usual Grüneisen relation, ${\mathrm{\alpha B}}_T{\mathrm{\hspace{0.17em}=\hspace{0.17em}\gamma C}}_{\upsilon }\mathrm{\hspace{0.17em}/\hspace{0.17em}V}$ , valid only for acoustical modes, is generalized for optical modes to give ${\mathrm{\alpha B}}_T{\mathrm{\hspace{0.17em}=\hspace{0.17em}\Sigma }}_j{\mathrm{(\gamma }}_j{\mathrm{\hspace{0.17em}/\hspace{0.17em}V)\; (\partial S}}_j{\mathrm{\hspace{0.17em}/\hspace{0.17em}\partial ln\omega }}_{\mathrm{jmax}})$ , where $S_j$ and ${\omega }_{\mathrm{jmax}}$ are the entropy and frequency cutoff of mode $j$ . The predicted thermodynamic properties of the system are unusually sensitive to chain length because of interchain thermal energy effects. The parameters are given values from independent solid n‐alkane data, and the calculated results are shown to predict quite well experimental $C_p$ and $\mathrm{V–T}$ behavior of the n‐alkanes.