Collection of Solved Problems in Physics

Diameter of Oleic Acid Molecule

Task number: 2079

• List of known information

  ρ = 900 kg m−3	oleic acid density
  26 % of total volume	spaces

  ---

  From tables:

  NA = 6.023·1023 mol−1	Avogadr’s constant
  Ar(C) = 12.011	relative carbon atom mass
  Ar(H) = 1.0079	relative hydrogen atom mass
  Ar(O) = 15.999	relative oxygen atom mass

• Analysis

  Avogadro’s constant tells us how many particles there are in one mole of matter. If we find out the volume of one mole of oleic acid, we can determine the volume of one molecule. Since we assume that their shape is spherical, we can calculate the volume easily from sphere volume equation.

  We must know the density to determine the molar volume of the oleic acid. The density is given and the molar mass can be found using the relative atom masses of those atoms that make up the oleic acid. All can be found in tables.

  We also have to consider that 26 % of the total volume are spaces between molecules, the molecules themselves have only 76 % of the molar volume.

• Hint 1 - molar mass

  Using the relative atom masses of carbon, hydrogen and oxygen that can all be found in tables, determine the molar mass M_m of the oleic acid.

• Molar mass of oleic acid

  We need to know the molar mass of oleic acid to calculate the diameter of its molecule. We will calculate it using the equation (derivation of the equation is in task Vztah mezi relativní molekulovou a molární hmotností(not translated yet)):

  \[M_\mathrm{m} = A_\mathrm{r}·10^{−3}\mathrm{kg mol^{−1}}\]

  First we will find the relative atom masses of carbon, hydrogen and oxygen. Then we will calculate the molar mass of all carbon, hydrogen and oxygen atoms in the oleic acid:

  \[M_\mathrm{m}(\mathrm{C}) = 18·A_\mathrm{r}(\mathrm{C})·10^{−3} = 18·12.011·10^{−3} = 216.198·10^{−3} \mathrm{kg mol^{−1}},\] \[M_\mathrm{m}(\mathrm{H}) = 34·A_\mathrm{r}(\mathrm{H})·10^{−3} = 34·1.0079·10^{−3} = 34.2686·10^{−3} \mathrm{kg mol^{−1}},\] \[M_\mathrm{m}(\mathrm{O}) =2·A_\mathrm{r}(\mathrm{O})·10^{−3} = 2·15.999·10^{−3} = 31.998·10^{−3} \mathrm{kg mol^{−1}}.\]

  Then we will sum the molar masses of carbon, hydrogen and oxygen and we will receive the total molar mass of the oleic acid:

  \[M_\mathrm{m}(\mathrm{C}_{17}\mathrm{H}_{33}\mathrm{COOH}) = 2.86·10^{−1} \mathrm{kg mol^{−1}}.\]

• Hint 2 - molar volume

  Formulate the molar volume of the oleic acid using its molar mass and density and from there formulate the volume of one molecule using Avogadro’s constant. Don’t forget to subtract the volume of the spaces between molecules.

• Hint solution

  Molar volume V_m is given by the formula

  \[V_\mathrm{m} = \frac{M_\mathrm{m}}{ρ},\tag{1}\]

  where M_m is the molar mass of the given matter and ρ is its density.

  Spaces between molecules make up 26 % of the molar volume, the molecules themselves make up only 76 % of the molar volume. If we divide this volume by Avogadro’s constant (we divide the volume of one mole by the amount of particles in one mole), we will receive the volume of one molecule V_O

  \[V_\mathrm{O} = \frac{0{,}74 V_\mathrm{m}}{N_\mathrm{A}}.\tag{2}\]

  The molecule has a spherical shape. Its volume is thus given by formula

  \[V_{O} = \frac{4}{3}πr^{3} = \frac{πd^{3}}{6},\]

  we express the diameter from the molecule’s volume

  \[d = \sqrt[3]{\frac{6V_O}{π}}\]

  and we will use the equations (2) and (1) with this equation

  \[d = \sqrt[3]{\frac{6·0.74V_\mathrm{m}}{πN_A}} = \sqrt[3]{\frac{6·0.74M_\mathrm{m}}{πρN_\mathrm{A}}}.\tag{3}\]

• Numerical substitution

  We will substitute the molar mass, that we have calculated, into equation (3), the density from the assignment and Avogadro’s constant:

  \[d = \sqrt[3]{\frac{6·0.74·M_\mathrm{m}}{πρN_\mathrm{A}}} \dot{=}\, \sqrt[3]{\frac{6·0.74·2.86·10^{−1}}{900π·6.022·10^{23}}} \mathrm{m}\] \[d \dot{=}\, 9.07 ·10^{−10} \mathrm{m}.\]

• Answer

  The diameter of an oleic acid molecule is approximately 9.07·10⁻¹⁰ m.

• Link to experiment

  An experiment to measure the molecule’s diameter can be found here: Diameter of oleic acid molecule.