The following is a crude but effective method for estimating...

The following is a crude but effective method for estimating the order of magnitude of Avogadro's number using stearic acid $\left(\mathrm{C}_{18} \mathrm{H}_{36} \mathrm{O}_{2}\right) .$ When stearic acid is added to water, its molecules collect at the surface and form a monolayer; that is, the layer is only one molecule thick. The cross-sectional area of each stearic acid molecule has been measured to be $0.21 \mathrm{nm}^{2} .$ In one experiment it is found that $1.4 \times 10^{-4} \mathrm{g}$ of stearic acid is needed to form a monolayer over water in a dish of diameter $20 \mathrm{cm}$. Based on these measurements, what is Avogadro's number? (The area of a circle of radius $r$ is $\pi r^{2}$.)

Key Concepts

Order-of-Magnitude Estimation

Order-of-magnitude estimation is a technique used to approximate the scale of a quantity, typically by rounding to the nearest power of ten. This approach is valuable in physics and chemistry because it simplifies complex calculations and provides a quick, reasonable estimation of quantities that span many orders of magnitude, such as Avogadro's number.

Surface Area Calculation

Surface area calculation involves determining the total area over a geometric shape, such as a circle, using mathematical formulas like ?r². This is crucial in experiments where the area covered by molecules is linked to the number of molecules present, facilitating estimations of molecular dimensions and densities on surfaces.

Monolayer Formation

Monolayer formation refers to the process by which molecules arrange themselves in a single layer on a surface. This concept is important in surface chemistry and materials science because it allows scientists to study molecular dimensions, spacing, and packing properties, which are essential for understanding interfacial phenomena and estimating molecular sizes.

Avogadro's Number

Avogadro's number is a fundamental constant that defines the number of constituent particles, usually atoms or molecules, found in one mole of any substance. It bridges the macroscopic measurements used in chemistry with the microscopic scale of individual particles, enabling calculations and conversions between the mass of a substance and the number of molecules present.

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