Ettevalmistus Keemiaolümpiaadiks/Kvantkeemia

Kvantkeemia

Kvantkeemia on keemia haru, milles probleemide lahendamiseks rakendatakse kvantmehaanikat.

Kvantmehaanika sai alguse Max Plancki aastal 1900 tähelepanekust, et musta keha kiirguse spektri seletamiseks peaks valgus tekkima ja neelduma diskreetsete "energiaportsjonitena" – kvantidena, mis erinevad teineteisest Plancki konstandi $h$) ja sageduse kordarvu võrra: $E=h\nu$, kus $\nu$ on sagedus. Kvantmehaanikas kohtab teisigi füüsikalisi suurusi, mis võtavad diskreetseid väärtusi. Oluliseks suuruseks on impulsimoment ($p$): $p=hk$, kus $k$ on lainevektor.

Põhimudelid

Osake 1D karbis mudel

$2L_{x}=n_{x}\lambda$

$p={\frac {h}{\lambda }}={\frac {h}{2}}{\frac {n_{x}}{L_{x}}}$

$E_{\mathrm {K} }={\frac {mv^{2}}{2}}={\frac {p^{2}}{2m}}$

$E_{\mathrm {K} }={\frac {h^{2}}{8m}}\left({\frac {n_{x}^{2}}{L_{x}^{2}}}\right)$

Osake 2D karbis mudel

$E_{\mathrm {K} }={\frac {h^{2}}{8m}}\left({\frac {n_{x}^{2}}{L_{x}^{2}}}+{\frac {n_{y}^{2}}{L_{y}^{2}}}\right)$

Osake 3D karbis mudel

$E_{\mathrm {K} }={\frac {h^{2}}{8m}}\left({\frac {n_{x}^{2}}{L_{x}^{2}}}+{\frac {n_{y}^{2}}{L_{y}^{2}}}+{\frac {n_{z}^{2}}{L_{z}^{2}}}\right)$

Osake ringil mudel

$2\pi R=l\lambda$

$p={\frac {h}{\lambda }}={\frac {h}{2\pi }}{\frac {l}{R}}$

$E_{\mathrm {K} }={\frac {mv^{2}}{2}}={\frac {p^{2}}{2m}}$

$E_{\mathrm {K} }={\frac {h^{2}}{8\pi ^{2}m}}\left({\frac {l^{2}}{R^{2}}}\right)$

Osake sfääril mudel

$E_{\mathrm {K} }={\frac {h^{2}}{8\pi ^{2}m}}\left({\frac {l(l+1)}{R^{2}}}\right)$

Viriaalteoreem

The vir­ial the­o­rem re­lates the ex­pec­ta­tion ki­netic en­ergy of a quan­tum sys­tem to the poten­tial. Let's con­sider a quan­tum sys­tem in a sta­tion­ary state, which does not have to be the group state. Let's assume that there is a single particle with position $q$ in a potential $V(q)$. The virial theorem relates the expectation kinetic energy $E_{\mathrm {K} }$ to the potential $V$ as follows:

$2E_{\mathrm {K} }={\Big \langle }q\cdot {\frac {\partial V}{\partial q}}{\Big \rangle }=nE_{\mathrm {P} }$

Harmooniline kvantostsillaator

For harminic osciallator $V={\frac {1}{2}}kx^{2}$, thus $2E_{\mathrm {K} }={\Big \langle }{\frac {1}{2}}x\cdot {\frac {\partial kx^{2}}{\partial x}}{\Big \rangle }=2V$. Then accoding to the virial theorem the expectation kinetic energy and the expectation potential energy are the same. The total energy is then $E_{\mathrm {T} }=E_{\mathrm {K} }+E_{\mathrm {P} }=2E_{\mathrm {K} }=2E_{\mathrm {P} }$.

Kineetiline energia avaldub kui $E_{\mathrm {K} }={\langle p^{2}\rangle }/{2\mu }$, kus ${\langle p^{2}\rangle }-{\langle p\rangle }^{2}={\langle \Delta p\rangle }^{2}$. Kuna ${\langle p\rangle }=0$, siis ${\langle p^{2}\rangle }={\langle \Delta p\rangle }^{2}$ ning $E_{\mathrm {K} }={\frac {{\langle \Delta p\rangle }^{2}}{2\mu }}$.

Potentsiaalne energia avaldub kui $E_{\mathrm {P} }=k\langle x^{2}\rangle$, kus ${\langle x^{2}\rangle }-{\langle x\rangle }^{2}={\langle \Delta x\rangle }^{2}$. Kuna ${\langle x\rangle }=0$, siis ${\langle x^{2}\rangle }={\langle \Delta x\rangle }^{2}$ ning $E_{\mathrm {P} }={\frac {k{\langle \Delta x\rangle }^{2}}{2}}$.

Väljundame $E_{\mathrm {T} }$ kui ${\sqrt {2E_{\mathrm {K} }\cdot 2E_{\mathrm {P} }}}$. Siis $E_{\mathrm {T} }={\sqrt {{\langle \Delta x\rangle }^{2}/{\mu }\cdot 2k{\langle \Delta x\rangle }^{2}}}={\langle \Delta p\rangle }{\langle \Delta x\rangle }{\sqrt {\frac {k}{\mu }}}$.

Vastavalt Heisenbergi määramatuse printsiibile, ${\langle \Delta p\rangle }{\langle \Delta x\rangle }\geq {\frac {h}{4\pi }}$, seega $E_{\mathrm {T} }\geq {\sqrt {\frac {k}{\mu }}}{\frac {h}{4\pi }}$, ehk $E_{\mathrm {T} }\geq {\frac {1}{2}}h\nu$, kus sagedus $\nu ={\frac {1}{2\pi }}{\sqrt {\frac {k}{\mu }}}$.

Harmoonilise kvantostsillaatori nullenergia võrdub $E_{0}={\frac {1}{2}}h\nu$. Kõrgemad energiad on üks teisest suurem $h\nu$ võrra.

$E_{n}=h\nu \left(n+{\frac {1}{2}}\right)$

Vesinikuaatomi Bohri mudel

Kineetiline energia avaldub kui $E_{\mathrm {K} }={p^{2}}/{2\mu }$, kus $p={\frac {h}{\lambda }}$. Kuna Bohri mudelis $2\pi r=n\lambda$, siis $E_{\mathrm {K} }={\frac {h^{2}n^{2}}{8\pi ^{2}r^{2}\mu }}$.

Potentsiaalne energia avaldub kui $E_{\mathrm {P} }=-{\frac {Ze^{2}}{4\pi \epsilon _{0}r}}$.

Vesinikuaatomi puhul Coulomb'i potentsiaal on $V=-{\frac {Ze^{2}}{4\pi \epsilon _{0}r}}$ ning $2E_{\mathrm {K} }={\Big \langle }-r\cdot {\frac {\partial }{\partial r}}{\frac {Ze^{2}}{4\pi \epsilon _{0}r}}{\Big \rangle }=-V$, ehk $2{\frac {h^{2}n^{2}}{8\pi ^{2}r^{2}\mu }}={\frac {Ze^{2}}{4\pi \epsilon _{0}r}}$.

Saame avaldada $r$ kui $r={\frac {\epsilon _{0}h^{2}}{Ze^{2}\mu }}n^{2}$.

Aatomi koguenergia on $E_{\mathrm {T} }=E_{\mathrm {K} }+E_{\mathrm {P} }=-E_{\mathrm {K} }={\frac {1}{2}}E_{\mathrm {P} }$.

$E_{n}=-{\frac {Z^{2}e^{4}\mu }{8\pi \epsilon _{0}^{2}h^{2}}}{\frac {1}{n^{2}}}$