91Ó°ÊÓ

The giant hornet Vespa mandarinia japonica preys on Japanese bees. However, if one of the hornets attempts to invade a beehive, several hundred of the bees quickly form a compact ball around the hornet to stop it. They don't sting, bite, crush, or suffocate it. Rather they overheat it by quickly raising their body temperatures from the normal \(35^{\circ} \mathrm{C}\) to \(47^{\circ} \mathrm{C}\) or \(48^{\circ} \mathrm{C}\) which is lethal to the hornet but not to the bees (Fig. 18-44). Assume the following: 500 bees form a ball of radius \(R=2.0 \mathrm{~cm}\) for a time \(t=20\) min, the primary loss of energy by the ball is by thermal radiation, the ball's surface has emissivity \(\varepsilon=0.80\), and the ball has a uniform temperature. On average, how much additional energy must each bee produce during the 20 min to maintain \(47^{\circ} \mathrm{C} ?\)

Step by step solution

01

Calculate the Surface Area of the Ball

Since the bees form a compact ball with a uniform temperature, we can model this as a sphere. The surface area of a sphere with radius \( R = 2.0 \) cm is given by the formula: \[ A = 4 \pi R^2 \]Substituting the given radius:\[ A = 4 \pi (0.02)^2 \] \[ A \approx 0.0050 \text{ m}^2 \] Thus, the surface area of the ball of bees is approximately 0.0050 m².

02

Use Stefan-Boltzmann Law to Calculate Power Loss

The power radiated by a surface due to thermal radiation is given by the Stefan-Boltzmann Law:\[ P = \varepsilon \sigma A T^4 \]where:- \( \varepsilon = 0.80 \) is the emissivity,- \( \sigma = 5.67 \times 10^{-8} \text{ W/m}^2 \cdot \text{K}^4 \) is the Stefan-Boltzmann constant,- \( A = 0.0050 \text{ m}^2 \) is the surface area,- \( T = 47 + 273.15 = 320.15 \text{ K} \) is the temperature in Kelvin.Plugging in the values:\[ P = 0.80 \times 5.67 \times 10^{-8} \times 0.0050 \times (320.15)^4 \]\[ P \approx 4.5 \text{ W} \]Thus, the power loss due to radiation is approximately 4.5 watts.

03

Calculate Total Energy Lost Over 20 Minutes

The total energy lost over a period of time \( t \) can be calculated using the formula:\[ E = P \cdot t \]where \( P \) is the power loss (4.5 W) and \( t = 20 \) minutes is the time, which should be converted to seconds: \[ t = 20 \cdot 60 = 1200 \text{ s} \].Substitute these values:\[ E = 4.5 \times 1200 \]\[ E = 5400 \text{ J} \]Thus, the total energy lost over 20 minutes is 5400 joules.

04

Calculate Energy per Bee

To find the additional energy each bee needs to produce, divide the total energy by the number of bees:\[ E_{\text{per bee}} = \frac{E}{N} \]where \( E = 5400 \text{ J} \) is the total energy and \( N = 500 \) is the number of bees.Substitute these values:\[ E_{\text{per bee}} = \frac{5400}{500} \]\[ E_{\text{per bee}} = 10.8 \text{ J} \]Each bee must produce approximately 10.8 joules of additional energy.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is vital for understanding how objects radiate energy due to their temperature. It states that the power radiated by a blackbody is proportional to the fourth power of its absolute temperature. This can be mathematically expressed as \( P = \varepsilon \sigma A T^4 \). Below is what each term means:

• \( P \) is the power radiated in watts (W).
• \( \varepsilon \) is the emissivity of the material, ranging from 0 (no emission) to 1 (perfect emission).
• \( \sigma \) is the Stefan-Boltzmann constant, approximately \( 5.67 \times 10^{-8} \, \text{W/m}^2 \cdot \text{K}^4 \).
• \( A \) is the surface area in square meters (m²) through which the radiation is emitted.
• \( T \) is the temperature in Kelvin (K).

This law helps us understand why the bees can effectively use their body heat. By forming a ball, they collectively increase their surface area and produce thermal radiation that weakens the hornet.

Surface Area Calculation

When bees form a ball to combat the hornet, they essentially create a small sphere. Calculating the surface area of this sphere is crucial in determining how much energy they can radiate. The surface area \( A \) of a sphere is calculated using:\[A = 4 \pi R^2\]
Here, \( R \) is the radius of the sphere. For the bees' ball, \( R = 2.0 \text{ cm} \), which we convert to meters (0.02 m) for consistency with other units.
Substituting this radius into the formula, we have:\[A = 4 \pi (0.02)^2 \]This results in approximately \( 0.0050 \text{ m}^2 \). A larger surface area would emit more energy, which is critical for the bees' strategy to raise the hornet's temperature through thermal radiation effectively.

Energy per Bee

Determining how much energy each bee must contribute in this situation involves calculating the energy loss due to thermal radiation and dividing it by the number of bees. After using the Stefan-Boltzmann Law to find the overall power loss at 4.5 watts, we find the total energy lost over the 20-minute time period:\[E = P \cdot t = 4.5 \times 1200 = 5400 \, \text{joules}\]
We then distribute this energy equally among the 500 bees:\[E_{\text{per bee}} = \frac{5400}{500} = 10.8 \, \text{joules}\]Thus, each bee needs to produce an additional 10.8 joules of energy to maintain the lethal temperature. This division of labor among the bees ensures the hornet is rendered inactive without endangering their hive.