# Solution to Q9 Let's solve each part step-by-step. --- ## a) **Average velocity of a Bucky ball at 25°C** ### **Given:** - Average kinetic energy = \( \frac{3kT}{2} \) - Mass of nanoparticle = \( m \) - Temperature \( T = 25^\circ \text{C} = 298 \text{ K} \) - \( C_{60} \) (Bucky ball) has molecular mass: \( 60 \times 12 = 720 \) amu - 1 amu \( = 1.66 \times 10^{-27} \) kg - Boltzmann constant \( k = 1.38 \times 10^{-23} \) J/K ### **Step 1: Calculate the mass of one Bucky ball** \[ m = 720 \times 1.66 \times 10^{-27} \ \text{kg} \] \[ m = 1.195 \times 10^{-24} \ \text{kg} \] --- ### **Step 2: Relate kinetic energy to velocity** The average translational kinetic energy is: \[ \frac{1}{2} m v^2 = \frac{3}{2} kT \] \[ v^2 = \frac{3kT}{m} \] \[ v = \sqrt{\frac{3kT}{m}} \] --- ### **Step 3: Plug in the values** \[ v = \sqrt{\frac{3 \times 1.38 \times 10^{-23} \times 298}{1.195 \times 10^{-24}}} \] \[ = \sqrt{\frac{1.23318 \times 10^{-20}}{1.195 \times 10^{-24}}} \] \[ = \sqrt{1.032 \times 10^{4}} \] \[ = 101.6 \ \text{m/s} \] **Final Answer (a):** > The average velocity of a Bucky ball at 25°C is **~102 m/s**. --- ## b) **Bucky ball in a vacuum micro-cell** ### **Scenario:** - Bucky ball in a glass micro-cell, at 25°C, but in vacuum (no other particles). - Ignore van der Waals (VDW) forces. - Will its average velocity be the same as above? --- ### **Explanation:** - In the first case, the Bucky ball is in thermal equilibrium with the medium (gas/liquid), so it has a thermal velocity as calculated above. - In the vacuum (second case), there are **no collisions** with other particles, so the Bucky ball cannot exchange energy with the environment. - If initially at rest, it will **remain at rest**; if given some initial velocity, it will continue at that velocity (no friction). - **Average velocity will NOT be the same as above** — it will be constant, determined by initial conditions, not by temperature. - **No Brownian motion** will be seen, since Brownian motion arises from collisions with molecules in the medium. --- ### **Trajectory Comparison:** #### - **In Gas/Liquid (Brownian motion):** - The Bucky ball’s path is a **random walk** due to continuous collisions with molecules. - Path: Jagged, erratic (see example sketch below). #### - **In Vacuum (Micro-cell):** - The Bucky ball travels in a **straight line** until it hits a wall, then bounces (elastic collision). - Path: Straight lines between wall bounces (see example sketch below). --- #### **Trajectory Sketch:** ```plaintext Brownian motion (Gas/Liquid): Vacuum (Micro-cell): _/\__/\_/\/\_/\/\_\_/ ------------------- / \_/ \_/ | | / /\_/ \/ /\ \ | \ / | \_/\/ \_/ \/ \_/ | \_______/ | | | ------------------- ``` --- ### **Summary Table** | Case | Average Velocity | Path/Trajectory | |--------------------|-----------------|---------------------------| | Gas/Liquid | ~102 m/s | Random, Brownian motion | | Vacuum (Micro-cell)| Undefined/Constant | Straight lines between walls | --- ## **Final Answers** ### **a)** - The average velocity of a Bucky ball at 25°C is **~102 m/s**. ### **b)** - In a vacuum, the Bucky ball will **not exhibit the same average velocity** as above; its velocity will be constant (set by initial conditions). - The motion will be **straight-line** between elastic collisions with the walls, unlike the **random Brownian motion** seen in gas/liquid. - **Trajectory:** Straight lines with bounces at the walls (not random walk).