Ch 20: The Micro/Macro Connection

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 20: The Micro/Macro Connection

Problem 46

Chapter 20, Problem 46

Dust particles are ≈ 10 μm in diameter. They are pulverized rock, with ρ ≈ 2500 kg/m³. If you treat dust as an ideal gas, what is the rms speed of a dust particle at 20℃?

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Convert the temperature from Celsius to Kelvin using the formula: \( T = T_{\text{Celsius}} + 273.15 \). For 20℃, \( T = 20 + 273.15 \).

The root-mean-square (rms) speed of a particle in an ideal gas is given by the formula: \( v_{\text{rms}} = \sqrt{\frac{3k_B T}{m}} \), where \( k_B \) is the Boltzmann constant \( (1.38 \times 10^{-23} \ \text{J/K}) \), \( T \) is the temperature in Kelvin, and \( m \) is the mass of a single particle.

Calculate the volume of a single dust particle assuming it is spherical. The formula for the volume of a sphere is \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius. The diameter is given as \( 10 \ \mu\text{m} \), so \( r = \frac{10}{2} \ \mu\text{m} = 5 \ \mu\text{m} = 5 \times 10^{-6} \ \text{m} \). Substitute \( r \) into the formula for \( V \).

Using the density \( \rho = 2500 \ \text{kg/m}^3 \), calculate the mass of a single dust particle with the formula \( m = \rho V \). Substitute the value of \( V \) from the previous step into this formula.

Substitute the values of \( k_B \), \( T \), and \( m \) into the formula for \( v_{\text{rms}} \) to calculate the root-mean-square speed of the dust particle.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas. It is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature in Kelvin. This law allows us to understand the behavior of gases under various conditions, which is essential for calculating properties like speed.

Root Mean Square (RMS) Speed

The root mean square speed is a statistical measure of the speed of particles in a gas. It is calculated using the formula v_rms = sqrt(3kT/m), where k is the Boltzmann constant, T is the absolute temperature, and m is the mass of a particle. This concept is crucial for determining the average speed of particles in an ideal gas, which helps in understanding their kinetic energy and behavior.

Kinetic Theory of Gases

The Kinetic Theory of Gases explains the macroscopic properties of gases in terms of the motion of their molecules. It posits that gas particles are in constant random motion and that their collisions with each other and with the walls of their container result in pressure and temperature. This theory underpins the derivation of the RMS speed and connects microscopic particle behavior to macroscopic gas properties.

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