Question

An experimentalist observed the motion of soot particles of radius 0.5 × 10−4 cm in
water-glycerine solution characterised by h = 2.80 × 10−3 kg m−1 s−1 at 300K for 10 s.
The observed value of D x2 was 3.30 × 10−8 cm2. Calculate Boltzmann constant and
hence Avogadro’s number.

Answer 1

An experimentalist observed the motion of soot particles of radius $0.5 \cdot 10^{-4} \, \text{cm}$ in water-glycerine solution characterized by $\eta = 2.80 \cdot 10^{-3} \, \text{kg} \cdot \text{m}^{-1} \cdot \text{s}^{-1}$ at $300 \, \text{K}$ for $10 \, \text{s}$. The observed value of $\overline{x^2}$ was $3.30 \cdot 10^{-8} \, \text{cm}^2$. Calculate Boltzmann constant and hence Avogadro's number.

Solution: According to the Einstein's theory of Brownian motion of molecules, main equation of which is:

$$\frac{\overline{x^2}}{2t} = \frac{k_B \cdot T}{6\pi \cdot \eta \cdot r}, \text{ then } k_B = \frac{3\pi \cdot \eta \cdot r \cdot \overline{x^2}}{t \cdot T} = \frac{3 \cdot 3.14 \cdot 2.8 \cdot 10^{-3} \cdot 5 \cdot 10^{-7} \cdot 3.3 \cdot 10^{-12}}{10 \cdot 300} = 1.45 \cdot 10^{-23} \, \frac{\text{J}}{\text{K}}; \text{ (all values for calculation were converted to the main SI units)}.$$

Avogadro's number can be calculated as: $N_A = \frac{\overline{R}}{k_B} = \frac{8.314}{1.45 \cdot 10^{-23}} = 5.73 \cdot 10^{23} \, \text{mol}^{-1}$; ($\overline{R}$ is the universal gas constant, 8.314 J·mol$^{-1}$·K$^{-1}$).

Answer: $k_B = 1.45 \cdot 10^{-23} \, \frac{\text{J}}{\text{K}}$; $N_A = 5.73 \cdot 10^{23} \, \text{mol}^{-1}$.