Smoke particles in the air typically have masses of the order of 10-16 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 rootmean-square speed of Brownian motion for a particle with a mass of 3.00 * 10-16 kg in air at 300 K.

Verified step by step guidance

1. Start by understanding the concept of Brownian motion, which is the random movement of particles suspended in a fluid (like air) resulting from their collision with fast-moving molecules in the fluid.
2. To find the root-mean-square (rms) speed of a particle, use the formula for the rms speed of gas particles: \( v_{rms} = \sqrt{\frac{3kT}{m}} \), where \( k \) is the Boltzmann constant \( (1.38 \times 10^{-23} \text{ J/K}) \), \( T \) is the temperature in Kelvin, and \( m \) is the mass of the particle.
3. Substitute the given values into the formula: \( m = 3.00 \times 10^{-16} \text{ kg} \) and \( T = 300 \text{ K} \).
4. Calculate the expression inside the square root: \( \frac{3 \times 1.38 \times 10^{-23} \times 300}{3.00 \times 10^{-16}} \).
5. Finally, take the square root of the result from the previous step to find the root-mean-square speed \( v_{rms} \).

Key Concepts

Brownian Motion

Brownian motion refers to the random movement of particles suspended in a fluid, resulting from collisions with fast-moving molecules in the fluid. This phenomenon is crucial for understanding the behavior of smoke particles in air, as it explains their rapid and irregular motion observed under a microscope.

Root-Mean-Square Speed

The root-mean-square speed is a statistical measure of the speed of particles in a gas, derived from the kinetic theory of gases. It provides an average speed value that accounts for the distribution of speeds among particles, and is calculated using the formula: v_rms = sqrt((3kT)/m), where k is the Boltzmann constant, T is the temperature, and m is the particle mass.

Kinetic Theory of Gases

The kinetic theory of gases explains the macroscopic properties of gases by considering their molecular composition and motion. It assumes that gas particles are in constant, random motion and that their collisions with each other and the walls of their container result in observable phenomena like pressure and temperature, which are essential for calculating the root-mean-square speed.