Oil stain test

Complete picture of the experimental setup

The oil stain experiment is an experiment from chemistry and physics that makes it possible to approximate both the size of an atom and the Avogadro number with simple means .

Historical

Benjamin Franklin made one such experiment on Mount Pond in Clapham Common and reported it in 1774. A teaspoon of oil spread over half an acre on the smooth lake (at a rate that surprised Franklin), resulting in a thickness of around 10 ⁻¹⁰ meters (a conclusion that Franklin did not make). At that time, Franklin was not yet interested in molecular dimensions (otherwise he would have had an estimate based on the assumption that the oil film was the thickness of a molecule), but like other scientists of his time in methods of smoothing out the waves of water with oil . His contemporaries, not even his friend Joseph Priestley , drew this conclusion; the idea of ​​molecular dimensions was not in the interests of scientists at the time, and Priestley himself even thought it possible that molecules were infinitely small. Franklin's contribution was soon forgotten, in Great Britain also because of his participation in the US War of Independence. Interest in it only reawakened at the end of the 19th century, when Lord Rayleigh (who knew Franklin's experiment) repeated it in public lectures at the Royal Institution in 1890 (also to demonstrate the smoothing of waves) and used it to estimate molecular dimensions in the Proceedings of the Royal Society. Further repetitions were published by Agnes Pockels in 1892 and Irving Langmuir in 1917.

Experimental setup

The oil stain in plan view and cross-section

Photo of an oil stain with a clear deviation from the round shape

A fine layer of bear moss spores - or a similar powder - is powdered onto a bowl filled with water . These serve to make the water surface more visible. Then a drop of a mixture of oleic acid and petroleum ether or mineral spirits with a known concentration and a previously determined volume is placed in the center of the bowl. The chemical formula of oleic acid must also be known. Petroleum ether or mineral spirits are used to distribute the oleic acid evenly and quickly over the water and to dilute the oleic acid, which is easier to dose in very small quantities. Petroleum ether or light petrol evaporates quickly, a stain of pure oleic acid remains on the surface of the water. This displaces the bear moss spores in a circle to the edge of the bowl, which is why it is clearly visible and can be measured with a ruler. Ideally, a perfect circle is created; mostly "frayed" however the edge.

evaluation

Of fundamental importance is the assumption that the oil stain is a monomolecular layer, i.e. that there are not several molecules on top of each other. This can be proven by simple additional experiments. If you add another oil drop of the same size, the area doubles exactly (i.e. the radius increases by the factor ). In addition, it is not possible to remove the stain z. B. to enlarge by blowing. ${\displaystyle {\sqrt {2}}\approx 1{,}4}$

Calculation of the molecular size

An oleic acid molecule in 3D; the experiment, however, uses the greatly simplified model of a cube

A spot that is as round as possible can now be viewed as a cylinder , with the diameter of a molecule corresponding to the height of the cylinder.

Using the volume of the drop and the concentration of petroleum ether (petrol) oleic acid, you can now calculate the volume of pure oleic acid, which is identical to that of the cylinder. With the measured radius you can now calculate the height of the cylinder. This is the size of an oleic acid molecule.

Calculating the atomic size

The oil stain test provides the approximate size of an atom with an atomic shell

The simplest form of calculation is based on cube-shaped molecules and atoms . Since one now knows the edge length, one can again calculate the volume of a molecule.

The chemical formula of oleic acid shows how many atoms a molecule consists of. It is now assumed that all atoms are the same size and completely fill the volume of the molecule. So if you divide the volume of a molecule by the number of atoms, you get the volume of an atom. Using a cube shape again, one can calculate the diameter of an atom. Depending on the accuracy of the implementation, an atomic radius of around meters is obtained, which is a very precise value given the simplicity of the experiment. ${\displaystyle 10^{-10}}$

Calculation of the Avogadro number

Now that you know the edge length of a molecule, you can calculate the number of molecules in the oil stain by dividing the volume of the oil by the volume of a molecule. From the definition of the Avogadro number , we know that the number of molecules in the oil stain is related to the Avogadro number as the mass of the oil in the oil stain is related to the mass of a mole of oil ( molar mass or molar mass), which can be derived from the chemical formula and the specifications of the Can be found in the periodic table . The result of the Avogadro number is also in the correct order of magnitude, depending on the experiment between about 5 · 10 ²³ and 7 · 10 ²³ , i.e. close to the real value of 6.022 · 10 ²³ .

Mathematical approach

Clarification of the symbols

Known or measured quantities:

• ${\displaystyle r_{\text{Fleck}}}$= Radius of the oil stain

• ${\displaystyle V_{\text{Tropfen}}}$= Volume of a drop

• ${\displaystyle V_{\text{Öl}}}$ = Volume of oil in one drop

• ${\displaystyle \rho _{\text{Öl}}}$= Density of the oil

• ${\displaystyle M_{\text{Öl}}}$= Molar mass of the oil

Intermediate results:

• ${\displaystyle d_{\text{Molekül}}}$= Height of the cylindrical oil spot and thus the diameter of an oleic acid molecule , in the model concept of cube-shaped molecules the edge length of a molecule

• ${\displaystyle V_{\text{Molekül}}}$ = Volume of one oleic acid molecule

• ${\displaystyle V_{\text{Atom}}}$= Volume of an atom

• ${\displaystyle d_{\text{Atom}}}$ = Diameter of an atom, in the model of cube-shaped atoms the length of the edge of an atom

• ${\displaystyle N}$ = Number of oil molecules in the oil stain

• ${\displaystyle m_{\text{Öl}}}$= Mass of the oil

Wanted sizes:

• ${\displaystyle r_{\text{Atom}}}$ = Radius of an atom

• ${\displaystyle N_{\text{A}}}$= Avogadro's constant

Required math formulas

• ${\displaystyle V=r^{2}\cdot \pi \cdot h}$ (Volume of a cylinder)

• ${\displaystyle h={\frac {V}{r^{2}\cdot \pi }}}$(Resolved after ) ${\displaystyle h}$

• ${\displaystyle V=a^{3}}$ (Volume of a cube)

• ${\displaystyle a={\sqrt[{3}]{V}}}$(Resolved after ) ${\displaystyle a}$

• ${\displaystyle {\text{Radius}}={\frac {\text{Durchmesser}}{2}}}$

• ${\displaystyle \rho ={\frac {m}{V}}}$

Calculation of the molecular size

${\displaystyle d_{\text{Molekül}}={\frac {V_{\text{Öl}}}{r_{\text{Fleck}}^{2}\cdot \pi }}}$

Calculating the atomic size

${\displaystyle V_{\text{Atom}}={\frac {V_{\text{Molekül}}}{\text{Anzahl der Atome}}}}$

${\displaystyle d_{\text{Atom}}={\sqrt[{3}]{\frac {d_{\text{Molekül}}^{3}}{\text{Anzahl der Atome}}}}}$

${\displaystyle r_{\text{Atom}}={\frac {1}{2}}{\sqrt[{3}]{\frac {\left({\frac {V_{\text{Öl}}}{r_{\text{Fleck}}^{2}\cdot \pi }}\right)^{3}}{\text{Anzahl der Atome}}}}}$

Calculation of the Avogadro number

${\displaystyle N={\frac {V_{\text{Öl}}}{V_{\text{Molekül}}}}}$

${\displaystyle N={\frac {V_{\text{Öl}}}{d_{\text{Molekül}}^{3}}}}$

${\displaystyle {\frac {N}{N_{\mathrm {A} }}}={\frac {m_{\text{Öl}}}{M_{\text{Öl}}}}}$

${\displaystyle N_{\mathrm {A} }=N\cdot {\frac {M_{\text{Öl}}}{m_{\text{Öl}}}}}$

${\displaystyle N_{\mathrm {A} }={\frac {V_{\text{Öl}}}{\left({\frac {V_{\text{Öl}}}{r_{\text{Fleck}}^{2}\cdot \pi }}\right)^{3}}}\cdot {\frac {M_{\text{Öl}}}{\rho _{\text{Öl}}\cdot V_{\text{Öl}}}}}$

${\displaystyle N_{\mathrm {A} }=\left({\frac {r_{\text{Fleck}}^{2}\cdot \pi }{V_{\text{Öl}}}}\right)^{3}\cdot {\frac {M_{\text{Öl}}}{\rho _{\text{Öl}}}}}$

Numerical example

Well-known sizes

• ${\displaystyle r_{\text{Fleck}}=6{,}4\,\mathrm {cm} =6{,}4\cdot 10^{-2}\,\mathrm {m} }$

• ${\displaystyle V_{\text{Tropfen}}=2{,}13\cdot 10^{-8}\,\mathrm {m} ^{3}}$( Standard drop gtt = 1/20 cm³)

• ${\displaystyle V_{\text{Öl}}={\frac {V_{\text{Tropfen}}}{2000}}=1{,}06\cdot 10^{-11}\,\mathrm {m} ^{3}}$ (because mixing ratio 1: 2000)

• ${\displaystyle \rho _{\text{Öl}}=900\,\mathrm {\frac {kg}{m^{3}}} =9{,}00\cdot 10^{2}\,\mathrm {\frac {kg}{m^{3}}} }$

Molar mass of oleic acid

The mass of a carbon atom is 12 u , that of an oxygen atom is 16 u and that of a hydrogen atom is 1 u. The formula for oleic acid C ₁₇ H ₃₃ COOH shows that a molecule consists of 18 carbon, 2 oxygen and 34 hydrogen atoms. The molecular mass is 282 u, which means that the molar mass of oleic acid is 282 g / mol.

invoice

${\displaystyle d_{\text{Molekül}}={\frac {1{,}06\cdot 10^{-11}\,\mathrm {m^{3}} }{(6{,}4\cdot 10^{-2}\,\mathrm {m} )^{2}\cdot \pi }}}$

${\displaystyle d_{\text{Molekül}}=8{,}24\cdot 10^{-10}\,\mathrm {m} }$

${\displaystyle d_{\text{Atom}}=2{,}18\cdot 10^{-10}\,\mathrm {m} }$

${\displaystyle r_{\text{Atom}}=1{,}10\cdot 10^{-10}\,\mathrm {m} }$

${\displaystyle N_{\mathrm {A} }=\left({\frac {6{,}4\cdot 10^{-2}\,\mathrm {m^{2}} \cdot \pi }{1{,}06\cdot 10^{-11}\,\mathrm {m} ^{3}}}\right)^{3}\cdot {\frac {2{,}82\cdot 10^{-1}\,\mathrm {kg} }{9{,}00\cdot 10^{2}\,\mathrm {\frac {kg}{m^{3}}} }}}$

${\displaystyle N_{\mathrm {A} }=5{,}6\cdot 10^{23}}$

Web links

• Oil stain test - test setup ( LEIFI )

References and comments

1. ^ Franklin, Philosophical Transactions of the Royal Society, Volume 64, 1774, p. 445
2. ↑ Franklin's experiment was repeated at the original location by Charles Giles. Giles, Chem. Ind., 1969, p. 1616. With photos.
3. ^ Charles Tanford, Ben Franklin Stilled the Waves: An Informal History of Pouring Oil on Water with reflections on the up and downs of scientific life in general, Oxford UP 2004, pp. 138f.
4. ↑ Strictly speaking, this is the way to measure the ratio of the mass unit gram to the atomic mass unit u . Since the reform of the units of measurement in 2019 , this quotient is no longer the Avogadro number by definition, but an independent number that can be determined experimentally. But the difference is significantly less than a billionth and can be ignored here.