# Introduction to Inorganic Chemistry/Basic Science of Nanomaterials

## 11.7 Problems

1. Consider a spherical gold nanoparticle that is 3 nm in diameter. If the diameter of an atom is approximately 3 Å, how many atoms are on the surface of the particle? What fraction of the gold atoms in the particle are on the surface?

2. Now consider a 3 nm diameter droplet of mercury. Mercury atoms are also about 3 Å in diameter. The heat of vaporization of bulk mercury is 64.0 kJ/mol, and the vapor pressure of mercury is 0.00185 torr = 2.43 x 10-6 atm. The surface tension of mercury (γHg) is 0.518 N/m, and the surface excess energy can be calculated as γHgA, where A is the surface area. Using this information and the Clausius-Clapyron equation (P = const•exp(-ΔHvap/RT)), calculate the vapor pressure of 3 nm droplets of mercury.

3. James Heath and coworkers (Phys. Rev. Lett. 1995, 75, 3466) have observed Ostwald ripening in thin films of gold nanoparticles at room temperature. Starting with an uneven distribution of particle sizes, they find that the large particles grow at the expense of smaller ones. Can you explain this observation, based on your answers to problems (1) and (2)?

4. The bandgap of bulk silicon is 1.1 eV. What bandgap would you expect for a 3 nm diameter Si nanocrystal? Use the Brus formula,

$\Delta {{E}_{gap}}\approx \frac{{{h}^{2}}}{8\mu {{d}^{2}}}-\frac{{{e}^{2}}}{4d\pi \varepsilon {{\varepsilon }_{0}}}+...$

where d is the particle diameter, εSi = 12, h = 6.6 10-34 J s, 1 eV = 1.6 10-19 J, 1 J = 1 kg m2/s2. and e2/4πε0 = 1.44 10-9 eV m. Assume that the electron-hole reduced mass μ is approximately equivalent to the electron mass, me = 9.1 10-31 kg.