International Chemistry Olympiad · 2006

Problems

Problem 1Avogadro's number. Spherical water droplets are dispersed in argon gas. At $27\ ^\circ\mathrm{C}$ , each droplet is 1.0 $\mu$ m in diameter and undergoes collisions with argon; assume inter-droplet collisions do not occur. The root-mean-square speed of these droplets was determined to be $0.50\ \mathrm{cm\ s^{-1}}$ at $27\ ^\circ\mathrm{C}$ . The density of a water droplet is 1.0 g cm $^{-3}$ . (1.1) Calculate the average kinetic energy ( $m v^2/2$ ) of this droplet at $27\ ^\circ\mathrm{C}$ . The volume of a sphere is $\frac{4}{3}\pi r^3$ . If the temperature changes, droplet size and speed also change. The average kinetic energy between $0\ ^\circ\mathrm{C}$ and $100\ ^\circ\mathrm{C}$ is linear in temperature; assume it remains linear below $0\ ^\circ\mathrm{C}$ and vanishes at absolute zero ( $0\ \mathrm{K} = -273\ ^\circ\mathrm{C}$ ). At thermal equilibrium, the average kinetic energy is the same irrespective of particle masses (equipartition theorem). The specific heat capacity at constant volume of argon gas (atomic weight 40) is $0.31\ \mathrm{J\ g^{-1}\ K^{-1}}$ . (1.2) Calculate Avogadro's number without using the ideal gas law, the gas constant, or Boltzmann's constant.Solutions: 1