Problem 33 - Physics

Question

Consult Interactive Solution \(\underline{14.33}\) at to see how this problem can be solved. Very fine smoke particles are suspended in air. The translational rms speed of a smoke particle is \(2.8 \times 10^{-3} \mathrm{~m} / \mathrm{s}\), and the temperature is \(301 \mathrm{~K}\). Find the mass of a particle.

Step-by-Step Solution

Answer

The mass of the smoke particle is approximately \( 1.59 \times 10^{-15} \) kg.

Step 1: Identify the Known Values

We are given the root mean square (rms) speed of the smoke particle, \( v_{rms} = 2.8 \times 10^{-3} \, \text{m/s} \), and the temperature, \( T = 301 \, \text{K} \). We are to find the mass of the particle.

Step 2: Recall the Formula for RMS Speed

The rms speed of a particle can be calculated using the formula: \[ v_{rms} = \sqrt{\frac{3kT}{m}} \]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.

Step 3: Rearrange the Formula to Solve for Mass

Reorganize the formula to solve for the mass \( m \) by squaring both sides and then isolating \( m \):\[ m = \frac{3kT}{v_{rms}^2} \]

Step 4: Substitute the Known Values

Plug the provided values into the rearranged formula:\[ m = \frac{3 \times 1.38 \times 10^{-23} \, \text{J/K} \times 301 \, \text{K}}{(2.8 \times 10^{-3} \, \text{m/s})^2} \]

Step 5: Calculate the Mass

Work through the calculations:\[ \begin{align*} m &= \frac{3 \times 1.38 \times 10^{-23} \times 301}{(2.8 \times 10^{-3})^2} \ &= \frac{1.24734 \times 10^{-20}}{7.84 \times 10^{-6}} \ &= 1.59138 \times 10^{-15} \, \text{kg}\end{align*} \]

Step 6: Conclusion

The mass of the smoke particle is approximately \( 1.59 \times 10^{-15} \, \text{kg} \).

Key Concepts

Root Mean Square Speed

The concept of Root Mean Square Speed (RMS speed) is crucial in understanding the kinetic theory of gases. RMS speed is a statistical average of the speed of particles in a gas. It provides a way to describe the speed distribution of particles, giving us a single, meaningful measure. The formula for RMS speed reflects this relationship, showing how speed is influenced by temperature and mass:\[ v_{rms} = \sqrt{\frac{3kT}{m}} \]Where:

• \( v_{rms} \) is the root mean square speed
• \( k \) is the Boltzmann constant
• \( T \) is the temperature in Kelvin
• \( m \) is the particle mass

The RMS speed increases with temperature, reflecting that particles move faster in hotter conditions. Similarly, it is inversely proportional to the particle's mass. This means lighter particles move faster than heavier ones at the same temperature.
Understanding RMS speed helps in linking the microscopic properties of particles to the macroscopic properties of gases.

Boltzmann Constant

The Boltzmann constant \( k \) is a fundamental constant that describes the relationship between temperature and energy at the particle level. It plays a crucial role in the kinetic theory of gases by bridging the gap between macroscopic and microscopic worlds. Its value is\[ k = 1.38 \times 10^{-23} \, \text{J/K} \]This constant appears in formulas that relate temperature to kinetic energy. It makes it possible to express the energy of gas particles in terms of temperature. In the formula for RMS speed, the Boltzmann constant directly influences how temperature translates to particle movement.

• Helps calculate thermal and kinetic energies
• Links macroscopic temperature to microscopic particle motion

Understanding the Boltzmann constant is essential for comprehending how particles behave under different thermal conditions and explains temperature's impact on gas behavior.

Translational Motion

Translational motion refers to the straight-line movement of particles, as they travel from one point to another. In the context of gases, particles are in constant translational motion, colliding with each other and container walls, contributing to gas pressure. This kind of motion is one of the three types of motion that gas particles can engage in, with the others being rotational and vibrational. RMS speed specifically measures translational motion, providing insight into how fast particles are moving on average. This measurement is essential for predicting the behavior of gas particles, such as how quickly they will spread out, interact, and create pressure.

• Occurs in three-dimensional space
• Responsible for diffusion and pressure in gases

Translational motion helps explain why gases occupy the volume of any container they're in, as particles continuously move and spread out to fill available space.

Particle Mass Calculation

Calculating the mass of a particle using given values, like RMS speed and temperature, involves rearranging the RMS formula to solve for mass. The restructured formula is:\[ m = \frac{3kT}{v_{rms}^2} \]This equation allows us to find the mass of an individual particle if we know its speed and the surrounding temperature. The calculation involves:

• Squaring the RMS speed
• Multiplying Boltzmann constant \( k \) by temperature \( T \)
• Dividing by squared RMS speed

As in the original solution, these steps provide numerical insights. For smoke particles suspended in air with an RMS speed of \( 2.8 \times 10^{-3} \, \text{m/s} \) and temperature 301 K, the resultant mass is approximately \( 1.59 \times 10^{-15} \, \text{kg} \). This concept shows the direct application of the kinetic theory principles to solve real-world physics problems.