One crude method of determining the size of a molecule is to treat the molecule as an infinite square well with an electron trapped inside, and to...

Monday, October 21, 2013

One crude method of determining the size of a molecule is to treat the molecule as an infinite square well with an electron trapped inside, and to...

Hello!

The formula seems correct. It can be obtained from the following facts:

$\displaystyle{E}_{{n}}=\frac{{{{n}}^{{2}}{{h}}^{{2}}}}{{{8}{m}{{L}}^{{2}}}}$ and $\displaystyle\lambda=\frac{{{h}{c}}}{{{E}_{{2}}-{E}_{{1}}}},$

where $\displaystyle{n}$ is the state number (changes from $\displaystyle{2}$ to $\displaystyle{1}$ in our case),
$\displaystyle{h}$ is the Planck's constant,
$\displaystyle{m}$ is the mass of electron,
$\displaystyle{L}$ is the size of a molecule,
$\displaystyle\lambda$ is the photon's wavelength.

Therefore $\displaystyle{E}_{{2}}-{E}_{{1}}=\frac{{{\left({{2}}^{{2}}-{{1}}^{{2}}\right)}{{h}}^{{2}}}}{{{8}{m}{{L}}^{{2}}}}=\frac{{{3}{{h}}^{{2}}}}{{{8}{m}{{L}}^{{2}}}},$ and

$\displaystyle{{L}}^{{2}}=\frac{{{3}{{h}}^{{2}}}}{{{8}{m}{\left({E}_{{2}}-{E}_{{1}}\right)}}}=\frac{{{3}{{h}}^{{2}}}}{{{8}{m}\frac{{{h}{c}}}{\lambda}}}=\frac{{{3}{h}\lambda}}{{{8}{m}{c}}}.$

Thus $\displaystyle{L}=\sqrt{{\frac{{{3}{h}\lambda}}{{{8}{m}{c}}}}}.$ To find the numerical result, recall the values in standard units:

$\displaystyle{h}={6.6}\cdot{{10}}^{{-{34}}},$ $\displaystyle{m}={9.1}\cdot{{10}}^{{-{31}}},$ $\displaystyle{c}={3}\cdot{{10}}^{{8}},$ $\displaystyle\lambda={1.94}\cdot{{10}}^{{-{6}}}.$

So the result is

$\displaystyle\sqrt{{\frac{{{3}\cdot{6.6}\cdot{1.94}}}{{{8}\cdot{9.1}\cdot{3}}}\cdot\frac{{{{10}}^{{-{34}}}\cdot{{10}}^{{-{6}}}}}{{{{10}}^{{-{31}}}\cdot{{10}}^{{8}}}}}}\approx\sqrt{{{0.176}\cdot{{10}}^{{-{17}}}}}=\sqrt{{{1.76}\cdot{{10}}^{{-{18}}}}}\approx{1.33}\cdot{{10}}^{{-{9}}}.$

This value is in standard units, meters. Because nano means $\displaystyle{{10}}^{{-{9}}},$ this is the same as $\displaystyle{1.33}{n}{m}.$ So your numerical answer is also correct:)

Ninenotes

Monday, October 21, 2013

One crude method of determining the size of a molecule is to treat the molecule as an infinite square well with an electron trapped inside, and to...

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