Detailed Study Notes for Lecture 17: Thermal Motion and Related Concepts

Lecture 17 Overview - Week 9: Thermal Motion and Related Topics

• Key Topics Covered:
  • Thermal Motion
  • Boltzmann’s Distribution
  • Perrin’s Experiment
  • Archimedes’ Principle
• References:
  • S.G. 16
  • Williams TEXT §13.1-13.3, 11.3
• Instructor: Prof. Rob "Boris" Wickham
• Office Hours: Mondays, 2:30 – 4:30 pm, MacN 448

Quiz and Assignments

• Quiz #4 covers SG 14 and 15.
• To master the material:
  • Read and complete SGs 14 and 15.
  • Online pretests due by Tuesday, November 11th: pretest #7 (SG 14), pretest #8 (SG 15).
  • Deadline for sign-up: Friday, November 14th.
• Upcoming Lab: Lab 4: Viscous Fluid Flow
• Due: Wednesday, November 5th.
• Don't Forget: Final Exam on December 1.

Thermal Motion of Molecules (SG16)

• Brownian Motion:
  • Described by Robert Brown in 1827. It is the spontaneous, irregular movement of microscopic particles suspended in a fluid.
  • It is the random motion of pollen grains resulting from collisions with unobservable, rapidly moving water molecules, demonstrating the existence and motion of molecules.
• Molecular Motion:
  • All molecules exhibit motion due to their temperature, denoted as T. This thermal energy is the kinetic energy of atoms and molecules within a system, which manifests as various types of motion.
  • Thermal energy can cause various types of motion:
  • Translational: Movement of the entire molecule from one point to another.
  • Vibrational: Oscillatory motion of atoms within a molecule around their equilibrium positions.
  • Rotational: Spinning motion of molecules around their center of mass.
  • Thermal energy represented as $k<em>B T$, where $k</em>B$ is Boltzmann’s constant defined as:
  • $k_B = 1.38 \times 10^{-23} \text{ J/K}$

Relationship between Temperature and Thermal Energy

• As an object heats up, its random motion increases; temperature is a direct measure of the average kinetic energy of the particles within the object.
• Examples of typical temperatures in Kelvin:
  • 0°C = 273 K
  • Room temperature: 21°C = 294 K \approx 300 K
• For room temperature, thermal energy is approximated as:
  • $k<em>B T</em>{\text{room}} \text{ approximately } 4 \times 10^{-21} \text{ J}$

Kinetic Energy of Particles

• Investigating movement of particles with thermal energy:
  • Kinetic energy at room temperature is substantially lower than that of human-scale objects, meaning thermal energy alone does not impart significant observable motion to large objects.
• Example for a baseball (mass = 150 g):
  • Resulting speed calculated using kinetic energy $\frac{1}{2}mv^2 = k<em>B T</em>{\text{room}}$ gives:
  • $v \text{ (speed)} \text{ approximately } 2.3 \times 10^{-10} \text{ m/s}$
  • Comparatively, a thrown fastball can reach speeds of 45 m/s. This highlights how negligible thermal motion is for macroscopic objects.

Importance of Thermal Energy for Smaller Objects

• Thermal energy is crucial for smaller particles (e.g., pollen, molecules); it is the dominant factor in their random motion.
• Example of a pollen grain:
  • Radius: 10 μm = $10^{-5} \text{ m}$
  • Density: 1.4 g/cm³ (converting to kg/m³: $1.4 \times 10^3 \text{ kg/m}^3$)
  • Velocities derived from thermal energy show:
  • $v \text{ (velocity)} \text{ approximately } 413 \text{ m/s}$
  • This high velocity indicates that thermal collisions profoundly affect smaller particles.

Distributions and Probabilities

• Height Distribution:
  • Not all individuals have the same height; it varies across a population.
  • Measurement example: Average height = 1.65 m from 800 students.
  • Probabilities can be assigned (e.g., what is the probability a person is 1.75 m tall?). This concept extends to molecular energies and positions.

Boltzmann’s Distribution

• For gas molecules, energies are not uniform but follow a distribution, with lower energy states being more probable.
• Boltzmann’s Equation:
  • The probability of a molecule having energy $E_1$ is given as:
  • $P(E<em>1) = \frac{N</em>1}{N<em>{\text{TOT}}} = C e^{\frac{-E</em>1}{k_B T}}$ where $C$ is a normalization constant to ensure the sum of probabilities equals one.
  • Similarly:
  • $P(E<em>2) = \frac{N</em>2}{N<em>{\text{TOT}}} = C e^{\frac{-E</em>2}{k_B T}}$
  • The ratio of probabilities (or number densities) between two energy states is:
  • $\frac{P(E<em>2)}{P(E</em>1)} = \frac{N<em>2}{N</em>1} = \frac{C e^{\frac{-E<em>2}{k</em>B T}}}{C e^{\frac{-E<em>1}{k</em>B T}}} \rightarrow \frac{P(E<em>2)}{P(E</em>1)} = e^{\frac{-(E<em>2 - E</em>1)}{k_B T}}$
  • This equation shows that the probability of finding a particle in a higher energy state decreases exponentially with increasing energy difference and decreasing temperature.

Applications of Boltzmann’s Distribution

• Barometric Formula:
  • Explanation of why all nitrogen molecules in the atmosphere do not sit at the same elevation, even if they have the same weight. Thermal motion counteracts gravity, leading to a distribution of particles at different heights.
  • The equations for gravitational potential energy at two different heights:
  • $E<em>1 = m g h</em>1$
  • $E<em>2 = m g h</em>2$
  • Ratios of number densities at different heights, incorporating buoyancy for particles in a fluid, can be formulated as:
  • $\frac{N<em>2}{N</em>1} = e^{\frac{-(\rho - \rho<em>f)Vg(h</em>2 - h<em>1)}{k</em>B T}}$ where $V$ is the particle volume, $\rho$ is the particle density, $\rho<em>f$ is the fluid density, $g$ is the acceleration due to gravity, and $(h</em>2 - h_1)$ is the difference in height.

Perrin’s Experiment

• Designed to test Brownian motion with particles in water, building upon Einstein's theoretical work.
• Included pollen grains in water to examine differences in density, specifically chosen to allow for observable distribution changes.
• Challenges: If particles are less dense than water, they float (buoyancy); if too dense, they sink (sedimentation). Finding the right density was key.
• Gamboge: Used as it was slightly denser than water and successful in demonstrating concepts of thermal motion and buoyancy. Perrin's experiments provided strong experimental evidence for the reality of atoms and molecules and directly confirmed Einstein's theory of Brownian motion. He used these observations to calculate an accurate value for Avogadro's number, providing a strong empirical basis for the kinetic theory of gases.

Archimedes’ Principle

• Definition:
  • When an object is placed in a fluid, it experiences an upward (buoyant) force equal to the weight of the fluid displaced. This force arises from the pressure difference exerted by the fluid on the object's top and bottom surfaces.
• Mathematical Representation:
  • Buoyant Force (B) = $\rho_f V g$
  • Weight of the object = $\rho V g$
  • Where $\rho$ = density of the object and $\rho_f$ = density of the fluid, V = volume, and $g$ is acceleration due to gravity.

Case Studies in Buoyancy

• Example Calculation:
  • Wooden ball in water with density 600 kg/m³ and radius 5 cm. Calculations involve comparing the weight of the ball to the buoyant force to determine if it floats or sinks and how much of it is submerged.
  • Adjust density values for steel (7800 kg/m³) and analyze force changes (gravity, buoyancy), illustrating how different densities affect an object's behavior in fluid.

Thought Experiments and Concept Questions

• Example of a practical problem involving buoyancy and stability of objects in liquid scenarios, such as calculating the maximum load a raft can carry.
• Scenario requiring assessment of maintaining buoyancy while carrying additional mass (e.g., treasure chest) in a raft. These questions often involve applying Archimedes' principle and understanding density relationships.

Analysis of Perrin’s Experiment

• Taking logarithms of both sides of Boltzmann’s equation helps analyze data relationships, turning exponential relationships into linear ones, making graphs and data fitting easier.
• The experiment demonstrated how well Boltzmann’s equation fits observed phenomena in terms of particle distribution in varying heights, providing quantitative confirmation of the theory.

Sample Problems

• Relative measurements at heights to analyze particle count under different conditions:
  • Task includes determining the number of particles at specific heights based on given data, using the barometric formula derived from Boltzmann's distribution.
• Assessment on how variations in particle volume affect ratios of particle counts at different elevations, emphasizing the role of effective mass in