Problem 70 - University Physics with Modern Physics

Question

Insect Collisions. A cubical cage 1.25 \(\mathrm{m}\) on each side contains 2500 angry bees, each flying randomly at 1.10 \(\mathrm{m} / \mathrm{s} .\) We can model these insects as spheres 1.50 \(\mathrm{cm}\) in diameter. On the average, (a) how far does a typical bee travel between collisions, (b) what is the average time between collisions, and (c) how many collisions per second does a bee make?

Step-by-Step Solution

Answer

(a) 0.02505 m, (b) 0.02277 s, (c) 43.92 collisions/s.

Step 1: Calculate the Volume of the Cage

The cage is a cube with each side measuring 1.25 m, so the volume is calculated using the formula for the volume of a cube, which is \( V = a^3 \), where \( a \) is the length of a side:\[ V = (1.25 \ \text{m})^3 = 1.953125 \ \text{m}^3. \]

Step 2: Determine Number Density of Bees

The number density \( n \) of the bees is the number of bees divided by the volume of the cage. With 2500 bees in the cage, we find:\[ n = \frac{2500}{1.953125 \ \text{m}^3} \approx 1280.0 \ \text{bees/m}^3. \]

Step 3: Calculate Effective Diameter of a Bee

Each bee is modeled as a sphere with a diameter of \( 1.5 \ \text{cm} \), which is \( 0.015 \ \text{m} \) in meters. Thus, the effective diameter \( d \) of a bee is \( 0.015 \ \text{m} \).

Step 4: Calculate Mean Free Path

The mean free path \( \lambda \) is given by the formula:\[ \lambda = \frac{1}{\sqrt{2} \pi d^2 n}. \]Substituting the values:\[ \lambda = \frac{1}{\sqrt{2} \times \pi \times (0.015 \ \text{m})^2 \times 1280.0 \ \text{bees/m}^3} \approx 0.02505 \ \text{m}. \]This is the average distance traveled by a bee between collisions.

Step 5: Calculate Average Time Between Collisions

To find the average time \( t \) between collisions, use the relation:\[ t = \frac{\lambda}{v}, \]where \( v = 1.10 \ \text{m/s} \) is the speed of a bee. Thus:\[ t = \frac{0.02505 \ \text{m}}{1.10 \ \text{m/s}} \approx 0.02277 \ \text{s}. \]

Step 6: Calculate Number of Collisions Per Second

The number of collisions per second is the reciprocal of the average time between collisions:\[ \text{Collisions per second} = \frac{1}{t} = \frac{1}{0.02277 \ \text{s}} \approx 43.92 \ \text{collisions/s}. \]

Key Concepts

Mean Free Path in Kinetic Theory of Gases

In the kinetic theory of gases, one fascinating concept is the mean free path, symbolized by \( \lambda \). This term describes the average distance a particle travels before colliding with another particle. To picture this with our bee problem, think of each bee as a small, spherical particle buzzing around in a certain volume. The mean free path helps us understand how far each bee travels on average before it bumps into another bee. In our bee exercise, we determined the mean free path using the formula \[ \lambda = \frac{1}{\sqrt{2} \pi d^2 n} \]where \( d \) is the diameter of the bee and \( n \) is the number density or the concentration of bees in a specific volume. This tells us that the mean free path is inversely proportional to how packed the space is and how big the bees are. The formula accounts for the effective cross-sectional area presented by each bee (a factor of \( \pi d^2 \)) and the overlap of flight paths, indicated by the \( \sqrt{2} \) factor. Hence, increasing the size of the bees or the bee population density decreases the mean free path, meaning more frequent collisions.

Understanding Collision Frequency

Collision frequency refers to how often each particle in a gas—like our bees—collides with other particles. It's a key concept in kinetic theory as it helps quantify the dynamic interactions between particles. In our exercise, the average time between collisions for each bee was calculated as approximately 0.02277 seconds which we then inverted to determine the collision frequency. By using the formula:\[ \text{Collision frequency} = \frac{1}{t} \]where \( t \) is the mean time between collisions, we establish how frequently bees experience these collisions. So the collision frequency for the bees is about 43.92 collisions per second. This high number illustrates that bees are constantly interacting with one another, highlighting the energetic, dynamic nature of our hypothetical enclosed bee world. The faster the bees fly or the denser the population, the higher the collision frequency becomes, showcasing the importance of these variables in collision dynamics.

Number Density and Its Role

Number density, denoted by \( n \), is a crucial metric in the context of gases. It defines the concentration of particles—be it gas molecules or bees—in a given volume. Specifically, it's calculated as the number of particles divided by the volume they occupy. This measure is vital as it directly influences both the mean free path and collision frequency of the particles.For our bees, the number density was calculated as roughly 1280.0 bees per cubic meter. This value signifies how populated the cage is with bees, which directly affects how often collisions between the bees occur. A higher number density implies that the particles are closely packed, leading to more frequent collisions and a shortened mean free path. Conversely, a lower number density results in fewer collisions and longer distances traveled before impact. Therefore, number density is fundamental in understanding how variables like pressure, volume, and temperature affect the behavior of particles in a gas-like system.