A circular lake 1.0 km in diameter is 10 m deep (Fig. 16.14). Solar energy is incident on the

A circular lake 1.0 km in diameter is 10 m deep (Fig....

A circular lake $1.0 \mathrm{~km}$ in diameter is $10 \mathrm{~m}$ deep (Fig. 16.14). Solar energy is incident on the lake at an average rate of $200 \mathrm{~W} / \mathrm{m}^{2}$. If the lake absorbs all this energy and does not exchange heat with its surroundings, how long will it take to warm from $10^{\circ} \mathrm{C}$ to $20^{\circ} \mathrm{C} ?$

Key Concepts

Specific Heat Capacity

Specific heat capacity is defined as the amount of energy required to raise the temperature of a unit mass of a substance by one degree Celsius. This property plays a key role in determining how much thermal energy is needed to change the temperature of a material, thereby connecting energy input to temperature changes.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed in an isolated system, only transformed from one form to another. This fundamental concept is used to equate the absorbed energy (from a source like solar radiation) with the increase in thermal energy that results in a temperature increase.

Solar Radiation

Solar radiation refers to the electromagnetic energy emitted by the sun, which arrives at the Earth in the form of light and heat. It is usually quantified as power per unit area, and understanding this concept is crucial when analyzing how much energy is available to be absorbed by a system or an object over time.

Energy Absorption

Energy absorption is the process by which a material or system takes in energy from its surroundings, leading to a rise in its internal energy, usually in the form of heat. This concept is essential for understanding how an external energy source, like solar radiation, can cause a temperature change in an object or medium.

Transcript

00:01 In this question, we are given a lake of diameter 1 km, and we want to find out how long does it take for the sun to increase the temperature of the lake by 10 degrees.

00:18 So the change of the temperature of the lake to be 10 degrees.

00:24 We are given the diameter, the diameter of the lake to be 1 km or 10 per 3 meters.

00:33 And we are given that the intensity of the sun is 200 watts per meter square.

00:42 So to work on this question, we will need the equation q, close to mct, delta t.

00:50 This will tell us how much energy is required to increase the temperature of the lake by 10 degrees, right? but to find out the total energy required, this we need the mass.

01:09 And the mass will depend on the volume that we have, the total volume of water.

01:16 And since we are given that it is 10 meters steep, we can assume it to be like a cylinder, like a cylinder that is 10 meters steep, with a diameter of 1 km, so the volume itself is equal to pi r square times h or pi over 4 d squared times h.

01:54 Substituting the values in should get 7 .8 -54 times 10 to power of 6.

02:04 Just cube.

02:05 And then we can convert to mass by multiplying by the density.

02:10 The density is approximately 1 ,000 kg per meter cube.

02:23 Right? and we should get 7 .854 times 10 to power of 9 kilograms.

02:36 Now that we have the mass, we have the mass...

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