91Ó°ÊÓ

Agnes Pockels \((1862-1935)\) was able to determine Avogadro's number using only a few household chemicals, in particular oleic acid, whose formula is \(\mathrm{C}_{18} \mathrm{H}_{34} \mathrm{O}_{2}\) (a) What is the molar mass of this acid? (b) The mass of one drop of oleic acid is \(2.3 \times 10^{-5} \mathrm{g}\) and the volume is $2.6 \times 10^{-5} \mathrm{cm}^{3} .$ How many moles of oleic acid are there in one drop? (c) Now all Pockels needed was to find the number of molecules of oleic acid. Luckily, when oleic acid is spread out on water, it lines up in a layer one molecule thick. If the base of the molecule of oleic acid is a square of side \(d\), the height of the molecule is known to be \(7 d .\) Pockels spread out one drop of oleic acid on some water, and measured the area to be \(70.0 \mathrm{cm}^{2}\) Using the volume and the area of oleic acid, what is \(d ?\) (d) If we assume that this film is one molecule thick, how many molecules of oleic acid are there in the drop? (e) What value does this give you for Avogadro's number?

Step by step solution

01

(a) Calculate the molar mass of oleic acid

To find the molar mass of oleic acid \(C_{18}H_{34}O_{2}\), we need to add up the molar mass of all the atoms in the formula: Molar mass of oleic acid = (18 * atomic mass of C) + (34 * atomic mass of H) + (2 * atomic mass of O) Molar mass of oleic acid = (18 * 12.01) + (34 * 1.01) + (2 * 16.00) = 282.5 g/mol

02

(b) Calculate the number of moles of oleic acid in one drop

We're given that the mass of one drop of oleic acid is \(2.3\times10^{-5}\) g. To find the number of moles, we'll use the formula: Moles = (mass) / (molar mass) Moles of oleic acid in one drop = \(\frac{2.3\times10^{-5} \,\text{g}}{282.5\, \text{g/mol}} = 8.14\times10^{-8}\,\text{mol}\)

03

(c) Determine the length of one side of the molecule base (d)

We're given that the volume of one drop of oleic acid is \(2.6\times10^{-5}\,\mathrm{cm}^3\) and the area covered by that drop is \(70.0\,\mathrm{cm}^2\). To find the length of one side of the base of the molecule (d), we can use the formula: Volume = Area * height Since the height is given as \(7 d\), we can write: \(2.6\times10^{-5}\,\mathrm{cm}^3 = 70.0\,\mathrm{cm}^2 * 7 d\) Solving for d: \(d = \frac{2.6\times10^{-5}\,\mathrm{cm}^3}{70.0\,\mathrm{cm}^2 * 7} = 5.33\times10^{-8}\,\mathrm{cm}\)

04

(d) Calculate the number of oleic acid molecules in one drop

Now that we have the length of one side of the base of the molecule (d), we can find the area of the base: Area of base = \(d^2 = (5.33\times10^{-8}\,\mathrm{cm})^2 = 2.84\times10^{-15}\,\mathrm{cm}^2\) Next, we'll divide the total area of the oleic acid layer (\(70.0\,\mathrm{cm}^2\)) by the area of one molecule base to find the number of molecules in one drop: Number of oleic acid molecules in one drop = \(\frac{70.0\,\mathrm{cm}^2}{2.84\times10^{-15}\,\mathrm{cm}^2} = 2.46\times10^{16}\) molecules

05

(e) Determine Avogadro's number

To find Avogadro's number, we'll use the formula: Number of molecules = (number of moles) * (Avogadro's number) Substituting the previously calculated values: \(2.46\times10^{16} \,\text{molecules} = 8.14\times10^{-8}\,\text{mol} * \text{Avogadro's number}\) Solving for Avogadro's number: Avogadro's number = \(\frac{2.46\times10^{16} \,\text{molecules}}{8.14\times10^{-8}\,\text{mol}} = 3.02\times10^{23}\,\text{molecules/mol}\)

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