Question

Arthur Clarke wrote an interesting short story called "A Slight Case of Sunstroke." Disgruntled football fans came to the stadium one day equipped with mirrors and were ready to barbecue the referee if he favored one team over the other. Imagine the referee to be a cylinder filled with water of mass $60.0\mathrm{kg}$ at ${35.0}^{\circ }\mathrm{C}$. Also imagine that this cylinder absorbs all the light reflected on it from 50,000 mirrors. If the heat capacity of water is $4.20\cdot {10}^3\mathrm{J}/\left({\mathrm{kg}}^{\circ }\mathrm{C}\right),$ how long will it take to raise the temperature of the water to $100.{}^{\circ }\mathrm{C}$ ? Assume that the Sun gives out $1.00\cdot {10}^3\mathrm{W}/{\mathrm{m}}^2,$ the dimensions of each mirror are $25.0\mathrm{cm}$ by $25.0\mathrm{cm},$ and the mirrors are held at an angle of ${45.0}^{\circ }$

Short answer

Based on the given information, it will take approximately 7.42 seconds to raise the temperature of the water in the cylinder to 100°C using the mirrors and solar energy.

Step-by-step solution

Calculate the total energy needed#End_tag#

First, we need to calculate the total energy needed to raise the temperature of the water to ${100}^{\circ }\mathrm{C}$. We can use the formula for the heat capacity to find this out: $Q=mc\mathrm{\Delta }T$ Where: $Q$ = Total energy needed (Joules) $m$ = Mass of the water ($60.0\mathrm{kg}$) $c$ = Heat capacity of water ($4.20\cdot {10}^3\mathrm{J}/\left({\mathrm{kg}}^{\circ }\mathrm{C}\right)$) $\mathrm{\Delta }T$ = Change in temperature (${100}^{\circ }\mathrm{C}-{35.0}^{\circ }\mathrm{C}$) $Q=60.0\mathrm{kg}\times 4.20\cdot {10}^3\mathrm{J}/\left({\mathrm{kg}}^{\circ }\mathrm{C}\right)\times ({100}^{\circ }\mathrm{C}-{35.0}^{\circ }\mathrm{C})$ $Q=60.0\mathrm{kg}\times 4.20\cdot {10}^3\mathrm{J}/\left({\mathrm{kg}}^{\circ }\mathrm{C}\right)\times ({65}^{\circ }\mathrm{C})$ $Q=16.38\cdot {10}^6\mathrm{J}$ So, the total energy needed to raise the temperature of the water to ${100}^{\circ }\mathrm{C}$ is $16.38\cdot {10}^6\mathrm{J}$.#End_tag#

Calculate the total power received from the mirrors#End_tag#

Next, we need to find the total power received by the cylinder per second from all the mirrors. We are given the dimensions of each mirror and the angle at which they are held, as well as the power output from the sun. First, calculate the area of each mirror: $A=lw$ Where: $A$ = Area (${\mathrm{m}}^2$) $l$ = Length ($0.25\mathrm{m}$) $w$ = Width ($0.25\mathrm{m}$) $A=0.25\mathrm{m}\times 0.25\mathrm{m}=0.0625{\mathrm{m}}^2$ Then, since we have 50,000 mirrors, the total area of all mirrors collectively will be: $A_{total}=n\times A$ Where: $n$ = Number of mirrors (50,000) $A_{total}=\mathrm{50,000}\times 0.0625{\mathrm{m}}^2=3125{\mathrm{m}}^2$ Now, we can find the total power received by the cylinder per second: $P_{total}=I\times A_{total}\times \cos \theta$ Where: $I$ = Intensity of sunlight ($1.00\cdot {10}^3\mathrm{W}/{\mathrm{m}}^2$) $\theta$ = Angle of the mirrors (${45}^{\circ }$) $P_{total}=1.00\cdot {10}^3\mathrm{W}/{\mathrm{m}}^2\times 3125{\mathrm{m}}^2\times \cos ({45}^{\circ })$ $P_{total}=1.00\cdot {10}^3\mathrm{W}/{\mathrm{m}}^2\times 3125{\mathrm{m}}^2\times 0.707$ $P_{total}=2.21\cdot {10}^6\mathrm{W}$ So the total power received per second by the cylinder from the mirrors is $2.21\cdot {10}^6\mathrm{W}$.#End_tag#

Calculate the time needed to raise the temperature#End_tag#

Finally, we can find the time needed to raise the temperature of the water to ${100}^{\circ }\mathrm{C}$ using the total energy needed and the total power received per second: $t=\frac{Q}{P_{total}}$ Where: $t$ = Time needed (seconds) $t=\frac{16.38\cdot {10}^6\mathrm{J}}{2.21\cdot {10}^6\mathrm{W}}$ $t\approx 7.42\mathrm{seconds}$ Therefore, it will take approximately 7.42 seconds to raise the temperature of the water in the cylinder to ${100}^{\circ }\mathrm{C}$.

Key concepts

Heat Capacity

Heat capacity is a fundamental concept in thermodynamics that measures the amount of heat energy required to raise the temperature of a substance by a specific amount. Expressed in units of Joules per degree Celsius (J/°C) for a given mass, it signifies the thermal mass of a material, which reflects its ability to store heat energy. In the example of the referee represented as a cylinder filled with water, the heat capacity of water is given as 4.20 x 10³ J/(kg·°C). This value indicates that for each kilogram of water, 4,200 Joules of heat energy are required to increase its temperature by one degree Celsius.

In practical terms, a higher heat capacity means it takes more energy to change a substance's temperature, making it a good buffer against temperature fluctuations. To calculate the total energy needed to heat the water in the cylinder, we use the formula
Q = mcΔT,
where Q is the total energy needed, m is the mass of water, c is the heat capacity, and ΔT is the change in temperature. This provides a straightforward method for understanding the energy transfer in heating processes.

Energy Transfer

Energy transfer is the process of energy moving from one place or object to another. It is an essential concept in thermodynamics, encompassing the ways energy is exchanged between systems and their environments. In the context of the exercise, the energy is transferred in the form of heat from the Sun to the water in the cylinder via the mirrors. The power of sunlight (known as solar irradiance) is given as 1.00 x 10³ W/m², which helps in determining the amount of energy each mirror reflects on the cylinder.

The energy transfer is crucial when calculating how long it will take for the water temperature to reach 100°C. After determining the heat capacity, we calculate the area of each mirror and the cumulative area of 50,000 mirrors to know the total power reflected towards the referee. This power, intensified by the number of mirrors and the mirrors' alignment, demonstrates the collective energy transfer capability of the mirrored assembly.

Power Intensity

Power intensity relates to the amount of power (energy per unit time) per unit area. It is a measure often used in the study of radiation energy, such as that emanating from the Sun. Power intensity can tell us how much energy an area receives per second, and it is a key factor in our exercise scenario. Given the solar irradiance is 1.00 x 10³ W/m², each square meter of the mirror surface reflects 1,000 watts of power from the Sun.

In our mirrored stadium scenario, this power intensity helps us calculate the total energy being focused on the unfortunate referee by accounting for the total reflective area and the angle at which sunlight strikes the mirrors. With the cosine of the angle factored in, because the light isn't hitting perpendicularly, we get the actual power intensity impacting the water-filled cylinder. The final step links power intensity to the time it takes for energy transfer to heat the cylinder's water to boiling point, demonstrating that power intensity plays a crucial role in determining the rate at which energy transfer occurs.