Smoke particles in the air typically have masses of the order of 10-10 kg. The Brownian motion (rapid, irregular movement) of these particles, resulting from collisions with air molecules, can be observed with a microscope. (a) Find the root-mean-square speed of Brownian motion for a particle with a mass of 3.00 X 10-16 kg in air at 300 K. (b) Would the root-mean-square speed be different if the particle were in hydrogen gas at the same temperature? Explain.
Solution 40E Step1 : We need to find the root mean square speed of the particle This is given using 1/2m(v ) = 3/2 K × T rms Here 23 K Boltzmann constant = 1.3806 × 10 T absolute temperature = 300 K v root mean square velocity 16 m mass of particle = 3.00 × 10 kg We need to find root mean square velocity hence rearrange the equation We get vrms = 3K×T m Substituting the values 23 vrms = 3×1.380616300 K 3.00×10kg 20 vrms = 1.242160 3.00×10kg 5 vrms = 4.1419 × 10 3 vrms = 6.4357 × 10 m/s = 0.64 cm/s Hence the root mean square velocity is obtained as 0.64 cm/s