Question

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 thenormal,$\mathbf{35}\mathbf{\text{°C\hspace{0.17em}}}\mathbf{}\mathbf{\text{\hspace{0.17em}to}}\mathbf{}\mathbf{47}\mathbf{\text{°C\hspace{0.17em}}}\mathbf{}\mathbf{\text{\hspace{0.17em}or\hspace{0.17em}}}\mathbf{4}\mathbf{\text{8°C}}$which is lethal to the hornet but not to the bees (Figure). Assume the following:$\mathbf{500}$bees form a ball ofradius$\mathit{R}\mathbf{=}\mathbf{}\mathbf{2}\mathbf{.0}\mathbf{}\mathbf{\text{cm}}$foratime,$\mathit{t}\mathbf{=}\mathbf{20}\mathbf{}\mathbf{\text{min}}$the primary loss of energy by the ball is by thermal radiation, the ball’s surface has emissivity$\mathit{\epsilon }\mathbf{=}\mathbf{0}\mathbf{.80}$,and the ball has a uniform temperature. On average, how much additional energy must each bee produce duringthe $\mathbf{20}\mathbf{}\mathit{m}\mathit{i}\mathit{n}$
tomaintain$\mathbf{47}\mathbf{\text{°C}}$
?

Short answer

$0.81\text{J}$The additional energy that must be produced by each bee during 20 min to maintain $47°\text{C}$is.

Step-by-step solution

Identification of given data 

i) The initial and final temperature of bees are:$T_i=35°\text{Cor308Kand}{T}_f=47°\text{Cor320K}$

ii) No. of bees are$n=50.$

iii) Radius of the ball is$R=2.0\text{cmor0.02m}.$

iv) Time taken by the bees to form a ball is$t=20\min \text{or}1200\text{sec}.$

v) Emissivity of the ball’s surface is$\epsilon =0.80$

Understanding the concept of thermal radiation

Thermal radiation is the amount of electromagnetic radiation emitted from a material that is due to the heat radiated by the material depending on its temperature. Stefan's law states that the energy radiated per second by the unit area of a black body is directly proportional to the fourth power of the thermodynamic temperature. We can find the rate at which energy is emitted by the bees using Stefan’s law. The additional energy which must be produced by each bee for 20 min to maintain can be calculated from the radiation rate using the corresponding relation.

Formula:

The rate at which the sphere emits thermal radiation,$P_{rad}=\sigma ϵA{T}^4$ …(i)

Where,$\sigma$is the Stefan–Boltzmann constant and is equal to,$(5.67\times {10}^{-8}{\text{W/m}}^{\text{2}}{\text{.K}}^4)$$ϵ$is the emissivity of the substance, $A$is the surface of the area of radiation,is the thermodynamic temperature.

The surface area of the sphere is given: $A=4\pi R^2$ …(ii)

Where,$R$is the radius of the sphere.

The energy transferred as heat by the body,$Q=P.t$ …(iii)

Where$P$, is the power radiated by the body, i$t$s the given time of radiation.

Determining the additional energy produced

Surface area of the ball is given by the equation (ii) as:

$\begin{array}{c}A=4\left(3.142\right){\left(0.02\text{m}\right)}^2\\ =0.005027{\text{m}}^{\text{2}}\end{array}$

The rate at which bees emit energy is given using the equation (i) and the above value of area as:

$\begin{array}{c}{P}_{rad}=(5.67\times {10}^{-8}{\text{W/m}}^{\text{2}}{\text{.K}}^4)\left(0.80\right)\left(0.005027{\text{m}}^2\right)({320}^4-{308}^4){\text{K}}^4\\ =0.339\text{W}\\ ~0.34\text{W}\end{array}$

Again, using equation (iii), the energy produced by each bee is given as follows:

$\begin{array}{c}\frac{Q}{N}=\frac{P_{rad}\times t}{N}\text{(ForNnumberofbees)}\\ =\frac{0.339\text{W}\times 1200\text{sec}}{500}\\ =0.8136\text{W-s}\\ ~0.81\text{J}\end{array}$

Therefore, the additional energy which must be produced by each bee during 20 min to maintain $47°\text{C}$is.$0.81\text{J}$$0.81\text{J}$