Smoke particles in the air typically have masses of the order of 10^-16 kg . The Brownian motion

step 1 Hi everyone this is the problem based on kinetic theory of the gas. Here it is given mass of smo

Smoke particles in the air typically have masses of the...

Smoke particles in the air typically have masses of the order of $10^{-16} \mathrm{kg} .$ 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 $3.00 \times 10^{-16} \mathrm{kg}$ in air at 300 $\mathrm{K}$ . (b) Would the root-mean-square speed be different if the particle were in hydrogen gas at the same temperature? Explain.

Key Concepts

Effect of the Surrounding Medium

When considering the motion of particles in a fluid or gas, the properties of the surrounding medium—such as the mass and speed of its molecules—can influence the dynamics of collisions. However, in the context of calculating the RMS speed of a Brownian particle based on its thermal energy, the temperature of the medium is the critical factor, while differences in the gas type (for example, air versus hydrogen) affect other aspects like collision frequency and mean free path but not the thermal equilibrium value determined by temperature.

Temperature and Thermal Energy

Temperature in the context of kinetic theory is a measure of the average kinetic energy of particles in a substance. It determines how vigorously the molecules move, which in turn affects phenomena like Brownian motion. At higher temperatures, particles possess greater kinetic energy, leading to faster movements, whereas at lower temperatures, the movements slow down.

Brownian Motion

Brownian motion refers to the irregular, random movement of microscopic particles suspended in a fluid (or gas) resulting from collisions with the much smaller, rapidly moving molecules of the surrounding medium. This concept is fundamental in understanding how macroscopic observable phenomena arise from the microscopic thermal motion of particles.

Root?Mean?Square Speed

The root-mean-square (RMS) speed is a statistical measure of the magnitude of velocity of particles in a system, usually calculated from the kinetic theory. It represents the square root of the average of the squares of individual speeds, linking the thermal energy of the system to particle speeds, and is often used to characterize the random motions within the system.

Kinetic Theory of Gases

The kinetic theory of gases provides a microscopic explanation of the macroscopic properties of gases by modeling them as a large number of small particles in constant, random motion. It connects parameters like temperature, pressure, and volume to the average kinetic energy and speed of the molecules, offering a framework to calculate quantities such as the RMS speed based on temperature and particle mass.

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